The screach is definitely a sign of a problem. It's typically caused by the belt slipping, and a slipping belt will effect the cold air output of the A/C.
The problem may be as simple as a worn or loose belt (simple).
It may also be the tensioner spring is getting weak or the noise is caused not by the belt, but one of the bearings in the two idler pulleys used on the tensioner (Still faily easy to service).
Worst case, it IS the compressor, and the added drag at start-up is causing the belt to slip.
Also, the compressor "cycling" for short bursts, is normal. In most (if not all) modern systems, the A/C is cycled as part of the defrost mode. It serves two purposes.
First, in DEFROST the cabin air is circulated through the A/C coil, and then the heater core. Passing the air over the cold A/C coil helps condense any moister out of the cabin air. It is then passed through the heater core to warm it up. The now warm, dry air is passed to the defrost ducts at the base of the windshield. This helps better evaporate the moister condencing on the windshield, and dramatically helps clear the window over the older systems that used only warm, moist air (circulated only through the heater core only).
Second, it is claimed that cycling the compressor for short bursts helps to prolong the life of the compressor and A/C system. By cycling the compressor, the seals, valves, compressor, and such are kept lubricated. After all, modern refrigerants serve as lubricants as well. In my '99 Dodge truck, I can hear the compressor cycle any time the fan is on, even in fresh-air mode. So far, I've yet to have any need to have the system serviced at all - It must work.
