Even though the little light goes out, sometimes it feels like the A/C is still on when I turn it off.

Is it programmed to come on when the temp knob is set too it's coolest setting, even if the A/C button is off?

No, but I found out on my 2004 Highlander that the AC will come on in any of the 2 defrost modes and is not stated in owner's manual. The AC will come on in those two modes and the AC light will not come on.

The A/C has to come on in any defrost mode because the A/C dries the air before it hit's the windshield. If it didn't, the defrost mode would fogg the windshield and cause very big problems. Try to set the temp knob higher so you get warm A/C air.