GBA and other emulators will need a BIOS to work, you can find more info on the retropie wiki
https://github.com/RetroPie/RetroPie-Se ... oy-Advance
Maybe you can try changing the emulator it uses for SNES? (when you select a game, you're shown a screen that says "press a button to configure", option 1 changes the default emulator. Use the start button in this menu to select items) I'm not 100% sure what would cause some games to not work.
Check the power switch is soldered down properly and that it's being properly moved over to the on position and not halfway or anything like that. Also, try shutting it down using the menus (press start in the main menu and scroll all the way to the bottom of the list) to see if shutting down that way has the same problem.