Can this unit be used just as a doorbell if connected to existing doorbell wiring?
I just want it to use it as a doorbell, but have it look like the real thing.
Yes. So long as you connect this device to your original doorbell wiring which sends a signal to your mechanical door chime causing it to ring. If using it with only the rechargeable battery (no wiring), it has no way to communicate with the mechanical chime. An alternative would be to connect it to an echo device, preferably an echo show. Any echo device will sound off when it receives a signal wirelessly as the ring doorbell and the echo is on the same WiFi network and same account. An echo show will give you the benefit of seeing who rang before you even get to the door and is an excellent convenience and security thing.
I have this model, but the battery has started dying and needs to be recharged after about 35 days. Since this is used, do you think I would have the same problem with one of these referbs?
35 days 
, you must not have a lot of traffic. I set perimeters on mine and still have to recharge) change batteries around 9 -12 days.
Compared to me your getting great battery life. You could also buy another battery. Having that second battery already charged is really convenient.
It is old and doesn’t have a removable battery. Thank you for letting me know that for some 35 days is good. When it was new, it didn’t need to be charged as often.