Would a MAP sensor cause my engine to go flat on power?

Would it cause the idle to be rough when the engine is cold?

I don't see your engine type -
On a 2.2
A bad MAP sensor should set a DTC code P0106, P0107, or P0108
MAP is used by the PCM to control fuel delivery and ignition timing.

Having said that though, on one of my 2.2's I did resently had a bad MAP that did not set a code. Engine was poor on fuel and a bit lacking on power. Put in a new MAP as there are several tests listed in the OEM manual to test MAP's ability to perform correctly - and good as new.