I would think this sounds like the compressor is going bad; likely the bearings. Since you describe it not being able to start after it has run. The metal may have heated up while it was running and created more drag that is risisting the initial start up. Then it is cooling down and the metal shrinks reducing the internal drag.
Another possibility is that it is over heating and, instead of reaching set temperature as you describe, it is stopping on internal overload, an internal switch on all motors less than 1HP. Figure out why the motor is overheating. this can be several factors; dirty filters, low refrigerant, restricted refrigerant, liquid lines going over 20 feet high, bad TXV equalizer, etc.