Do I still get a free satellite if I skip the tutorial?

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.

When I played through the tutorial on Classic, I started with one satellite, and was prompted to launch another (that I did not have to pay for) as part of the tutorial mission.

When I later played while skipping the tutorial on Normal, the Situation Room still showed me with a "free" satellite.

So as far as you can tell, you don't get a "Free" satellite by going through the tutorial. (At least, on the first 2 difficulties)