Intercept from different bases possible in X-Com Enethe Unknown

It is in fact an event where you will get a chance to get more resources. This is really important; you need to acquire certain materials to do certain research to get a path to victory in the game, and that says nothing of getting your soldiers properly outfitted.

As to the randomness, that's not likely. The aliens have an agenda and it will play out on the geoscape. UFOs have missions; shooting them down prevents them from carrying those missions out. Allowing the aliens to carry out those missions will help spread panic/alien presence (at a high level) which is bad for you.

I have only ever seen ufos where I had satellites deployed. Eventually you will need to upgrade interceptors as they will start encountering much larger, more powerful craft.

You get a free satellite either by going through the tutorial or by just playing on Normal. You don't get one on Classic and I imagine you don't get one on Impossible either.