Sounds like a simple clean and/or replace of the IAC (Idle Air Control) valve may solve your problem.
I take mine off, douse it with some TB cleaner, followed by a liberal spray of silicone lubricant, & re-install.
As far as the vibration, you should feel a vibration for just an second when the A/C compressor clutch engages. Its just a slight drain on the engine & a proper IAC valve will adjust to it in just a few seconds.
