For example if SPY reached a day high of +2% but only for minutes, would call prices rise accordingly such that one could profit from the STC sale of an option of the same strike price?
Regarding sensitivity, securities with whose options have high Open Interest and activity are very sensitive to price change in the underlying. If you follow some quotes in real time, you'll see the moment to moment price change in both.
It's important to note that each option has its own delta so the responsiveness of option price is tied to the size of the delta. An OTM call with a delta of 20 is going to move 20 cents for a dollar move in the underlying. If you're dealing with an illiquid option with a wide spread, sometimes you might not see any price change in the option if the underlying moves 10 or 20, even 30 cents.
As for the second question about STC after a 2% rise in the SPY, it's not clear what your position is or what you are attempting to do.