We work with a lot worse at our local Bike Hospital... no reason why your bike can't be fixed to work right.

Chain path should be straight (wheel centrered in frame, chain ring not bent) chain tension should be fairly tight - looks a bit slack in your pic.

BMX chain is simple single speed, new ones can be as cheap as a pint of beer but I've only ever changed them for cosmetics voluntarily. Even the most wornout ones still stay on their tracks until they break.

Suggest installing a calliper/cable/lever front brake - not a fan of relying only on rear coaster on any kid bike.