Was the motor replaced with the same size (horsepower, amperage, voltage) motor? What is the amperage and voltage listed on the motor's data plate? In many cases the motor plate will either list the size of the required breaker, or if not, the manufacturers website will have that information available. If you "had the motor changed" it might be best to call the company that changed it and report that the new motor is tripping the breaker, where the old one did not, and have them check for issues, since they should have verified that the new motor would work the same, or made any changes needed to support any change in the motor. If you changed it yourself, that's on you.
You should not replace the breaker with a larger breaker unless the new motor requires a larger breaker.
You should not replace the breaker with a larger breaker without ALSO replacing the wire with larger wire, suited to the size of the larger breaker, which should be suited to the size of the new motor. 20 amp, 12 gauge copper, minimum - 30 amp, 10 gauge copper, minimum.
Rather than "sanding the wires" you should (with the breaker off, of course) cut off the damaged ends of the wire, pull it a little further in, and strip the ends to get new undamaged wire. If this is not possible, consider replacing the wire, especially if it's fairly short run.
110V (single-pole breaker?) is rather abnormal for most well pump installations. One potential fix would be to re-wire for 220V operation if possible
A well can allow oxygen into the soil where it will thin out any organic materials present, possibly causing subsidence. Without redigging the well using expensive methods it is hard to prevent this. Also, if the water is in a reservoir of some kind, reduction of that reservoir can cause subsidence. There is not much you can do about either of these problems except trying to locate the well elsewhere maybe.
Best Answer
Not really.
You can do that if you only turn on the pump when you're consuming water. You don't want the pump running if no water is moving. You'll wear it out.
The point of a pressure switch is to prevent just that, and the point of the pressure tank is to reduce the frequency of pump startups (along with stabilizing pressure during usage).