It's normal for the A/C to to load the engine down, but it should not cycle that short unless it's fairly cold out. You are probably low on freon.
The idle should not change that much. Either the IAC is not responding well, or your idle mixture is near the limit.
If you have had no CEL's, you could try cleaning the IAC, MAF, and fuel injectors. Make sure the secondaries are closed at idle. Make sure the PCV is good and there are no disconnected or leaking hoses on the intake system.
