Can I fit 26 x 2.2 tyres to my 26 x 2.25 wheels on my mountain bike?

If you want to put 2.2" tires in place of current 2.25" tires, then yes, you can do that.

That's less than a 2mm difference, given that tires conform to marketed width at all (which they tend not to). Generally, any smaller tire size (of compatible beat seat diameter) won't cause any issues, unless there's a huge mismatch between rim and tire width (i.e. 40mm internal width rims and 25mm wide tires aren't a good idea, same goes for 10cm fat tires and 18mm rims, but that's not your case).

I advise to consult this page on a topic of tires sizing.

You will be absolutely fine to make a change like this. Your tires are wear items and should be replaced when they are worn.

It depends on the rim width mostly, but you could probably go any tire from 1.75 up to 2.3 without problem.

For your usage you want smaller nobs not large ones. If 95+% of your ride is road then get some with a slick commuter strip around the middle, will help you go faster on the smooth and will have only minor difference off-road.