I quit reading after a bit but the title to the property next door has absolutely nothing to do with adverse possession of even prescriptive easement.
You do not have to own an adjacent property to claim adverse possesion. You merely have to adversely possess a piece of property, ANY property, for the prescribed period of time while fulfilling the other statutory requirements.
As well, if there was permission given to use the area in question, then that is not adverse possession. It is use with permission and does not count towards any time for an adverse possession claim. It actually stops am accumulation of time and restarts the clock.
actually, that is one of the easiest ways to prevent an adverse possession claim. Give formal permission to use the area. Then, along with permission granted, it can be as easily revoked.