It depends on which panel you have installed. Some can be taken over/repurposed and some cannot. Post a pic if you can

Yes, but you’ll have to replace the whole set. Check out Yale

i’m assuming you mean wired internet and not wired power, right? because you can get A) wired power and wifi, b) wired power & wired internet (POE), or C) battery + wifi. If your wifi isn’t great by the door, I would see if you can get by with an extender before you run cable. if running cable isn’t a big deal, that will be a much better end result.

Do you mean wired power or internet? If you can get both, that’s going to be the most stable… but most of the battery powered and wifi doorbells are fine. If you have strong wifi and don’t mind charging every few months, you can pretty much go with whatever has the features you want and not worry about connectivity.