Will a rotator cuff impingment or tear heal on its own, and will it cause arthritis in the shoulder joint?

Depends. Rotator cuff impingement is often a self limiting problemthat can resolve either on its own or with treatment. The rotator cuff is present these do not feel on their own but if our small can be treated like impingement. Larger rotator cuff tearsleft untreated can lead to arthritis of the show.
Complex answer. Impingment /rotator cuff tendinitis can heal given appropriate treatment. Tears generally do not heal but may become asympomatic. Untreated large tear may develop arthritis in the shoulder.