Wow, you miiight want to see if you can get a replacement.

I've had this happen to me with my first Tiny Arcade, you might want to return yours.
First, though, check if the bootloader text shows. (hold both buttons down and turn it on)

Is the world in the game procedurally generated or predefined?
Also, how can I compile the source code of the game? I'm really interested in trying this out.

If this is OK, then this thread can be where users can post ideas for games and such.

Some ideas from me:
Battery level indicator in the menu (where you select games)

Berzerk-like game (or just a clone of berzerk, it doesn't matter)
Tiny-Man (pac-man game, i'm bad with names)

Does it need a .bin file? Just the .tsv? What is the converter you used?

Only turns on in bootloader mode, worked fine until I tried to upload code to it (CORRECT PROGRAMMER, CORRECT PORT, CORRECT BOARDS, CORRECT EVERYTHING), then it completely shut down, and after that it only turned on if both buttons were pressed down.
Reloaded the game menu, works again.

Yes, it did. This post was actually for a defective model, a new one was sent to me and it works fine.

Does the orange light turn off after the Tiny Arcade is finished charging?