JoeViking245

Edit the game Click Additional Apps Select the one you want as the default play-when-you-double-click-it [in the lower-right] click Make Default Now, if you want to change the Names (titles), you can.  But that won't affect which is the default game to launch.

The 3D Box Model and the 3D Box Image are 2 separate things.  One can never directly replace the other.
The 3D Box Image is not one of the image types that will display in the game Details (right) panel.  Regardless of disabling the 3D Box Model.
In LaunchBox, the only places to visually see the 3D Box Image are:
When you edit the game When you double-click one the of smaller images in the games Details panel (to make it show full screen) press the left or right arrow on the keyboard until you see that image Select Image Group and change it to 3D Boxes This will change the main (middle) section images to 3D Boxes (Image)

When you say "category", I'm hoping you mean "Platform" because that'd make it a lot easier.
 
To move them to a different Platform:
When in the Super Nintendo Entertainment System Platform, Select the games that you want moved and then press Ctrl+E to start the Bulk Edit Wizard.  For the Field, select from the dropdown, Platform.  For the Value, select (or you can manually type it in) Nintendo Super Famicom (or however you want to word and spell it).  Follow through the prompts to complete the process.

K.  Then have it not check for updates for that emulator.
Tools < Manage < Emulators  - Click on your emulator then right-click on the Status column (for that emulator) and select Don't check for updates.

That error message is from FinalBurn Neo. Not LaunchBox.
The emulator needs to know where your ROMs are located.  I encountered an issue when I set the path(s) [in the emulator] then closed it, they weren't being saved.  So be sure to click Save settings (step 4). 
Open FinalBurn Neo Go to Misc < Configure ROM Paths... Add the path(s) to where your ROMs are located Let it finish scanning Go to Misc Click Save settings to .ini now Close the emulator  
In LaunchBox, edit your FinalBurn Neo emulator Check the box Remove file extension and folder path UN-CHECK the boxes Remove Quotes (quotes are needed if using ROMS that have spaces in the filename) Extract ROM archives before running Click OK to save and close  
Play games

Looking at the Change Log, the multiple instances of LaunchBox/BigBox had some 'safety checks' added in version 5.3.
 

The script you show is in the Running Script section of the emulator. (An educated guess)
To exit from the Pause Menu, that portion of the script [also] goes in the Exit Script section - 'The script to use when exiting from the Pause Menu'.
So in the emulators Exit Script section, put:
$Esc:: { if WinExist("JoyToKey") Process, Close, joytokey.exe Run, "D:\HFSBox\Utilities\JoyToKey\JoyToKey.exe" "Layout 8 Boutons.cfg" WinClose, ahk_exe {{{StartupEXE}}} } (while keeping the full script in the Running Script section.)
 
Note: I moved the 'close the emulator' to the last line.  I'm kind of surprise that when exiting from NOT in the pause menu that the script changes the profile back.
I say this because when the game is exited, the Running Script is essentially discarded/thrown-away/process-killed.  No matter where it's at.  It may be that the processes you have, take place "super-fast".  i.e. before the game is 'completely exited'.  That said, if it ain't broke, don't fix it.  

If you have autosave enabled in MAME itself (autosave 1), you don't need the -autosave parameter.  Adding that parameter just overrides whatever you have set in MAME.  Just like as if you executed MAME form the Windows command prompt.  
Same thing with "state_directory   sta" and the -state_directory <path> parameter.
 
If in LaunchBox you have Upload Your MAME High Scores... enabled (Tools < Options < Integrations < MAME), there's a caveat...
I know high scores and autosaves are 2 separate things.   But if it affects that, it's possible it has an effect on auto saves as well.  (I don't know for certain)

Edit you MAME emulator in LaunchBox and check the Command-line Parameters for -rompath %romlocation%.  If it is there, since you edited you own 'rompath' in mame.ini, remove that parameter and value.

When importing, the importer ignores everything inside parenthesis.  So probably the easiest way to keep your special numbers in the filenames would be to use a bulk file-rename utility and have it add parentheses around the number and then import those files with the 'new' filename. 
The 'new' filename would be for example:  (04.) Sonic The Hedgehog.ext

That's a good sign.  Try adding a 3 second timer before Process, Wait...  Giving the com commands a few moments to settle in.
If it still doesn't work, move the 1st 2 com commands into their own script and add that as an Additional App for the games and set them to Run Before Main App.  Try it on one or two and if it works, there's a plugin where you can bulk add Additional Apps to all the games [that need it].

Remove -fullscreen from the emulators Default Command-line Parameters (1st screenshot).  (It's not an available parameter [anymore?].)
Then, open [just] the emulator.  Go to Video - Display Configuration - and check Emulate Fullscreen and Fullscreen on ROM Open.

You have an extra - (dash) right before the "path\to\ROM.zip".  Remove it.
Run, %MAMERun% -"D:\LaunchBox Arcade\Games\Arcade\chasehq.zip", , Hide
Run, %MAMERun% "D:\LaunchBox Arcade\Games\Arcade\chasehq.zip", , Hide ; OR Run, %MAMERun% "%GameRDP%", , Hide  

Close LaunchBox.  (Click on it and press Alt+F4) Open Windows Explorer and navigate to //LaunchBox/Data/ Open Settings.xml with any text editor Find the line starting with <ControlsBarHeight> Set the value to 50 (is probably something like 1550, or some big number) <ControlsBarHeight>50</ControlsBarHeight>  
DO NOT TOUCH OR CHANGE ANYTHING OTHER THAN THAT NUMBER Save and close the file Restart LaunchBox Don't do what you did before to get it like that.

The Sample files go in the samples folder.
D:\Emulators\MAME\samples\seawolf.zip
Note the sample file and the ROM file have the exact same name. So don't get them mixed up.  

My cab's shut down for the night so can't look... but should box: be in the command-line parameters?  I do recognize --escape-exit though.

Hopefully it regenerated those too.
 
The notification is part of the new emulator installer/updater plugin feature in v13.15.
Go to Tools - Manage - Plugins.  Select the Dolphin plugin and click Update.  Then LaunchBox will stop asking.
It's not asking you to update your Dolphin installation. Just the plugin.

Since you set your ROM paths in MAME, try removing from the command-line parameters  -rompath %romlocation%

Sounds like what you want is a Custom Badge.
 

Try adding start "" to the beginning.  (And pretty sure remove the @ symbol also [for asterix.bat])
start "" ..\..\Emulators\hypseus\hypseus.exe singe vldp -framefile singe\asterix\asterix.txt -script singe\asterix\asterix.singe -sound_buffer 2048 -fullscreen -gamepad -scanlines -scanline_shunt 3 (in Cyber Shadow, add it in front of the last line... start "" "%emulator_path%".....)
 
 
I assume since you have the full path to the emulator, the batch file is not located in the same folder as the emulator.  That said you may need to switch to that folder, first.
cd ..\..\Emulators\hypseus start "" ..\..\Emulators\hypseus\hypseus.exe singe vldp -framefile singe\asterix\asterix.txt -script singe\asterix\asterix.singe -sound_buffer 2048 -fullscreen -gamepad -scanlines -scanline_shunt 3  
 
Then if you want to make use of LaunchBox's Startup and Shutdown screens for that game (have the Shutdown screen actually show after the game exits), add a wait parameter (/w) to the start line. (don't forget the "cd" line)
start /w "" ..\..\Emulators\hypseus\hypseus.exe singe vldp -framefile singe\asterix\asterix.txt -script singe\asterix\asterix.singe -sound_buffer 2048 -fullscreen -gamepad -scanlines -scanline_shunt 3 You'll then need to do the above shortcut thing to hide the batch file command window.  
 
...which now that I think about it, those screens may have been the issue.  If you're so inclined, don't do anything I've said to do, above.   Edit one of the games and turn off the Startup/Shutdown screens.  Startup/Pause - check Override Default Start Screen Settings.  Click OK to save and close.  If that fixes it, you can then do the wait/shortcut thingy and re-enable the Startup screen (un-check override) [if you want to have them show].

In Tools - Manage - Plugins, is the Dolphin plugin up-to-date (v1.2)?

Usually when a simple plugin like this doesn't work it's because it's blocked. But if you used 7-Zip to extract it, it should unblock it for you.  Double check by right-clicking the dll file, select properties and see if there's an option to unblock it. If there isn't, then it's ok.
I tried running LB as admin (which we should NEVER do).  But the plugin still showed.
Then I got a crazy idea.... I added a manifest.json file to the test plugins folder. Though the plugin showed up in Manage Plugins (not that you can do anything with it there), it would not appear on right-click.  So if you still have that json file in with the plugin, remove it.  Note to self: the manifest.json file is only for plugins that are internally managed by LaunchBox.
 
If it's not the json file thing, then I don't know.  I suppose maybe try a known working right-click-game plugin and see if it works.  Here's a small, simple one known to work.  Note in the instructions it talks about checking that it's unblocked.

I was referring to your MAME emulator settings in LaunchBox itself.  (Tools - Manage - Emulators - [select MAME] - Edit) 
No matter how you have things set in mame.ini, if you pass a parameter(s) via the command-line, it will take precedence over what's in any of the MAME configuration files.

Yes.  Update the plugin.
The notification is part of the new emulator installer/updater plugin feature in v13.15.
Go to Tools - Manage - Plugins.  Select the plugin needing updated and click Update.  Then LaunchBox will stop asking.
It's not asking you to update your emulator installation. Just the plugin.

The \LaunchBox\Core\Images\ folder is a symbolic link that only exists when LaunchBox/BigBox is running.  
The \LaunchBox\Images\Cache-BB\ folder (the one that's linked to when running) will change (update).  If the purpose of your syncing is just to maintain a backup of your build, I'd exclude that cache folder (also the Cache-LB folder) from the sync operation.
 
May look at setting up the sync software to only sync files that have changed.  I.e. set it for 'update" rather than "mirror".  That should save a significant amount of time.