JoeViking245

Like @C-Beats, I don't recall how the MAME installer was actually set up (compared to other less 'odd' or unique emulators).  But with other emulators, when you see a message similar to that, it will also ask if you want "us" to install it.
That aside.... Since you have your 30-40 classic Arcade games added to your LaunchBox collection already, at this point I'd suggest:
Go to Tools - Manage - Emulators and click Add Type in "MAME" for the Emulator Name and it will auto-populate most of the settings At the right-end of the Name field will be an icon that is a link to the MAME download site.  Click it. If your MAME ROMs are from the current 0.281 release, download the mame0281b_x64.exe file If your ROMs are from an earlier release, at the top of that page, click Downloads and select Previous Releases Scroll down to where you see the release number that matches your ROMs Click the file under Binary (Windows) for that release Once downloaded, run the executable from Windows Explorer If you get a Windows protected your PC popup, click More Info and then click Run anyway I recommend extracting it to your /LaunchBox/Emulators/ folder e.g. Choose:  D:\LaunchBox\Emulators\MAME\ Be sure to add MAME on the end of the path Once finished extracting, back in your Edit Emulator window To the right of Application Path, click Browse and navigate to the folder you extracted MAME to and select "mame.exe" Right below Default Command-line Parameters, add to the end of what's there (if it's not there already), -rompath %romlocation% e.g.  -keyboardprovider dinput -rompath %romlocation% Click on the Associated Platforms label and make sure Default Emulator is checked on the Arcade line. Click OK to save and close the Edit window.

Try.... under Tools < Options < Visuals < Game Details, un-check:
Possible Ports Recommended Games Similar Games

Going to Tools < Download < Force Update Games Database Metadata... should resolve the issue of missing games.

Uncheck Root.
In your last image, the upper arrow is for the Consoles (Category) checkbox. The lower arrow is for the Root checkbox.

Read the output on the screen?  
When you ran Check ROM(s), it showed you in red that some files are missing from within the zip(s) and what their names are.
This is most likely due to a mismatched ROM set and MAME executable.
Case-in-point:  Your Robocop ROM is for version 0.130 (or older).  The Winning Run ROM is for version 0.201 (or older).  (Don't ask how I found that out because it's a PITA. lol)
 
Solution: Get a ROM set that matches you MAME executable version (e.g. both are for version 0.251).  (That, I can not help you with)
Alternately, remove the game(s) from LaunchBox that won't play because of a mismatched set. (But that's no fun.)
 
Reading the onscreen output:  (e.g. Winning Run)
In the upper right, you'll see that a Device file is required for this game.  When running Check ROM(s), you'll see in the output 
Checking Device file...
sys2mcpu.bin CRC is good
But right above that, under Checking ROM file... was nvram is missing from the ROM winrun.zip.

Using the LaunchBox Install MAME (emulator) I think installs it into /LaunchBox/Emulators/MAME 0.2xx/.  Where 0.2xx is the version installed.  And if my recollection is correct, a 'new' version/release will get installed into a new folder (MAME 0.2xy). (this may have been changed since I last checked quite-a-while-ago).
Either way, you will NOT lose any custom configs.  A new MAME installation will only have the base files of the app, leaving all custom changes/additions/configs intact when installed over the top of an existing one.  (Unless you're a power user and have gone in and modified something like crt-geom.json)
When you say "it DOES import correctly", I assume you mean "install" [the emulator] correctly.  (vs. "import" the ROMs correctly?)
 
The afore mentioned updating procedure will [also] not change/delete the artwork (if you're talking MAME's /artwork/ folder [aka bezels]) nor will it mess with any images or videos you've downloaded in LaunchBox for your Arcade platform.  But I don't think you have the latter because of the issue with the MAME Arcade Fullset Import Wizard.
RE: updating your MAME set, get the 0.279 (actually, may as well get 0.280) fullset.  Essentially replacing your 0.261 set.  (As opposed to updating 0.261 to 0.262 to 0.263 etc. which is how it'd need to be done if you wanted to update your physical existing set. And is a PITA not a fun task.)
As for updating the emulator, if you're happy with the one LB installed for you, you can just copy from your existing MAME install the folders: artwork (if you have bezels in it), cfg, ctrlr (if you created a custom controller.cfg file. Not to be confused with the game-specific ones in the cfg folder),  ini
If you made custom changes to MAME (like ROM path), you probably have a mame.ini file that should be copied over as well.
 
IF you tried running the MAME Import Wizard and pointed to your 0.261 MAME install (accidentally or on purpose), that could be the problem.  Some things were changed in how the xml that gets produced [quite] a while back.  I forget when exactly.  But Nov. 2023 (0.261) in MAME terms, to me falls under "quite a while back".  

No.  
But there is a plugin you can use to rename the media once downloaded.

Select all the games in the platform (click one, then press Ctrl+A).
Tools < File Management < Change ROMs Folder Path for Selected Games...

You can’t hide it because it’s a security feature.
You can email support and ask them to change it if you’re streaming or something similar and have a personal security issue with your real name being displayed.

Short version:  Yes.
Long version:
[generally] LaunchBox does not keep 'record' (say, via an xml file or otherwise) of images associated with a game.  On-the-fly, it looks in the respective platforms folder under /Images/.  It looks for an image file matches the games Title or the games ROM filename (minus extension). (not sure of the order-of-precedence)
Note RE: game Title and image file names.  If the game Title has invalid filename characters, they will get replaced with an underscore when saving/naming the image file.
e.g.  Title = 19XX: The War Against Destiny  Image filename = 19XX_ The War Against Destiny.png
Additionally, LaunchBox also replaces single-quotes with an underscore.
 
That is correct.
 
Also correct.  Following the guidelines of "SelectedGame".
 
You're on a roll.   Coded in the Default Pause Theme (specific for "Arcade") for this image is SelectedGame.ArcadeControlsInformationImagePath
 
When you look in the /LaunchBox/PauseThemes/Default/ folder, you'll see Default.xml and Arcade.xml.  The latter being the (specific for "Arcade") mentioned above.
Default.xml is the Pause screen shown for platforms that don't have their own xml file.
Default.xml shows the images:
SelectedGame.ClearLogoImagePath SelectedGame.FrontImagePath SelectedGame.CartFrontImagePath So no control panel images.
 
Sort of.  No.
For Arcade[.xml], it dynamically shows the static control panel image for that SelectedGame (if one exists).  You're not that lucky to have available dynamic images like what CPWizard does.   In my image above, that panel was created using CPWizard, customized to match my cab (green and buttons etc. layout).  I then had CPW save an image file for each game. (Yeah, a ton of images. lol)
When you download images, (for Arcade) you can get (if available)
Arcade - Controls Information (a generic layout)

And/or Arade - Control Panel

 
You're welcome.  Now that you have a good basis, I think it's time to stop planning, start setting things up and get to playin'. 

You can set Region Priorities under Tools < Options < Data < Region Priorities.  And I believe that will use that region as the priority when selecting the Title during import. But TBH, I'm not for certain.
Since (it seems) you have your games already imported, there are a couple options (aside from re-importing).
Change each game manually Edit the game.  Go to the Alternate Names section. Click (select) the Name you wish to see. Click Set Selected Name as Title. Bulk change the game Titles to an Alternate Name found for a selected Region using this plugin:  

Email support@unbrokensoftware.com to get it resolved.  All licensing questions/concerns need to (can only) go through there.

Didn't watch the video to see what all it covered.  But this article from the LaunchBox Strategy Guide may help also.  (It covers emulators and their portability, required dependencies and maybe some other goodies.)
Moving Your LaunchBox Instance to a New PC - LaunchBox

Not really sure what to make of that, but I'll go with 'if you went to YouTube and played a video, you'd hear its audio.'  
I asked about the MAME version because know there were some audio issues with release 0.278.
The only reason MAME would stop having audio output is if a setting was changed (in MAME) or your speaker volume is turned down or muted, or your speakers were turned off. Since you indicated your sound is working fine with everything else, I'm inclined to say it's a setting in MAME itself.

The only thing I can help with is sharing my process.
My test:
Installed a fresh copy of LaunchBox 13.22 Closed the initial Add Games popup I don't think Download Bezels options popup via this wizard (could be wrong) Though they can always be added later via Download - Metadata and Media Tools - Manage - Emulators - Add RetroArch - Download OK - Close  
Tools - Import - ROM Files - Single Platform selected my Atari 2600 folder, followed default prompts Checked Download Bezels (selected default Theme Bezels) finished the process to import Selected Adventures of Tron and clicked Play.

The only thing in RetroArch I did go in and manually change was Settings - Input - and set Confirm Quit to Off.  Otherwise, nothing in RA was changed/setup/modified/etc.
The bezels are found in \LaunchBox\Emulators\RetroArch\overlays\GameBezels\Atari2600\[.png and .cfg files]

Select a game, then press Ctrl+A (to select all games in that platform).  Then Tools < Download < Update Metadata and Media for Selected Games...
Sticking with MAME (Arcade), I suggest selecting Advertisement Flyer - Front in leu of Box-Front as arcade cabinets weren't shipped in 'boxes'.  Crates, yeah.  
Videos and Manuals only come from EmuMovies.  The LaunchBox Database doesn't have those.  When you purchase you LB license, I believe they also provide a discount code for an EmuMovies subscription (which is required for downloading from within LB).
 
Again, sticking with MAME, yes.  It's easiest to manage if your emulator(s) are installed in /LaunchBox/Emulators/.  Then it's just a matter of moving the /LaunchBox/ folder.
Some emulators, by default are not portable. They'll store some of their data in your C:\Users\JoeViking\AppData\emu_name\ folder. (there, or in \Documents\, or somewhere else).  But most can be made "portable" so their data is stored within the emulators folder.  Varies by emulator.
 
Regarding bezels for other consoles that should have them, that will be console specific.  (aka need more info)  But that's all outside of my wheelhouse.  So the "more info" would be for someone else. 😁

They were both found here.  Make sure your Platform is set to Windows.  May also try to force update your local database.
Tools < Download < Force Update Games Database Metadata...
 


 
 

Edit your PacMan game. Under Emulation, check the box Use Custom Command-line Parameters. in the space below it, add -plugin pactrainer.
This will actually send the command mame -plugin pactrainer pacman. But should work just the same. 
If you use the LaunchBox Pause Menu for Arcade, shame on you.  But if that's your thing, you'll need to add -keyboardprovider dinput -plugin pactrainer for it to work.

Give this a read. Hopefully something in there will resolve the issue.
Plugin Access Denied Error - LaunchBox

The default location change in version 13.21 with the advent of Media Packs.  You can find more information about them here:
LaunchBox Media Pack Feature – Version 13.21 - LaunchBox

(I'm guessing you're meaning /LaunchBox/Emulators/RetroArch/)  No. It's not. 
If for some reason it truly is missing on yours, then I'm surprised RA even opens. As in it's probably a corrupt install.  If that's the case, have LaunchBox reinstall it for you in Tools - Manage Emulators. 
Otherwise, right-click a game using RA and follow the red boxes...

Update your local database
Tools < Download < Force Update Games Database Metadata...

Anything's possible.
To test, you could try removing the contents of your emulators \UserData\ folder.  Then copy the contents of your \AppData\Roaming\BigPEmu\ folder into it.  Then use the %romfile% -localdata parameter(s).
If that works, you can remove the /AppData/Roaming/BigPEmu/ folder.

Clear Logo folder location changed as of version 13.21.  Can read more about it here:
LaunchBox Media Pack Feature – Version 13.21 - LaunchBox

If you had LaunchBox install MAME for you, there should be some scripts in the Reset, Save and Load sections.
 
...and if those scripts had been in there, you could look at the one for Save, where it takes a modifier key and another keypress to activate.  Modify that script to suit your needs and you'd get
Send {Alt down} Sleep 50 Send {r down} Sleep 50 Send {r up} Sleep 50 Send {Alt up} The key here (no pun intended) is setting a Key Delay which is needed for some emulators.  Essentially, you're causing a more pronounced key press(es).
I haven't tested this in Supermodel because like any Arcade game, I don't save.  We call it "old school".   But I suspect it will work.