Hi and welcome to the community.
This actually sounds more like a emulator issue rather than a Launchbox one. You say your controllers are setup and detected, and even have different colours denoting that they are detected as two different controllers, but that player two does not work once in a game. This is a emulator issue as Launchbox has no effect on controllers once a emulator loads, it's only used for the Launchbox/Bigbox ui which the loads the emulator which takes over the controllers at that point. So i would suggest loading your emulator by itself and checking the input section, it may only have a player 1 controls set at the moment and will need player 2 controls set as well.
As for your nes/snes games not working also with your controller, again that sounds like a emulator problem, have you opened the emulator by itself and set the controls before you tried to launch through Launchbox/Bigbox?
And finally for 3DO, i have never personally emulated that so can be of little help there, but if it is simply not loading anything at all then i would check that the games load OK through just the emulator first, if they do then you must have a Launchbox configuration issue which we will need to troubleshoot. If they don't load through just the emulator itself either then you have a bigger problem. like incorrect bios, bad rips, wrong format rips ect.
Hope that helps you in some way.