Yes a 2.0 bottom end can be fitted with a 1.8 head. The gain in power is modest though. About 5 or 6 bhp gain I think.
About your gearbox. A couple of options. A longer gearbox from a MKIII will so straight in. You will lose a little bit of the acceloration sparkle but have a more relaxed cruiser instead of a screaming roller skate. Or do what I did. Go to a scrappy and find a Golf 1.6 CL with a four speed +E, and remove the fifth gear. When removing the 5th gear, remember how you did it, with taking measurements too, so that you know how to change it for the one in your gearbox. You will lose about 800RPM in 5th, over the original RPM.
So you retain the GTI performance in 1st, 2nd, 3rd and 4th, but have a relaxed motorway cruiser in 5th.
Rub jonny will have more info on gearbox selection and engine block interchangeability.