Ring 3 (battery) + Ring Wired - and do I need the jumper?

I have a Ring 3 (battery powered, but also now connected to the doorbell wires) and a more recent Ring Wired. I’m assuming it’s ok to run both on the same doorbell transformer, right? Second question - I’ve noticed that the Ring 3 battery shows wired on the app, but doesn’t every really charge. And I eventually have to pull the battery out and hook to a charger. That being said… I never used the jumper on the old doorbell. Is that why the Ring 3 isn’t changing?  Third question - I have an old chime. Wires are labeled 0, 1, 2, 3. Not sure what to jumper! (Pic added in comments).