Have you tried wiring it directly into the panel on a short bit of wire? Also check the fuses,fuses are there to save the panel when amatures decide to play around.
1. I wouldn't install something purchased by a customer, it opens up all sorts of warranty issues.
2.This would be my recommendation
3. All of this stuff is one way wireless, as swb says not an alarm system. The Ring alarm will need to be more reliable than the doorbell to be any good. It really comes down to what type of burglar you are likely to attract.