Declined assaulting the Overseer UFO after shooting it down. Any consequences

Well, I guess it qualifies as a minigame, but there's not really much to it until research + engineering opens up more options. As it stands, you close in on the enemy and automatically start attacking it. If you think your interceptor is going to get destroyed you can order it to withdraw, and that's it.

Via research, you can unlock one-time use power-ups that give you what are basically different options during the fight. They must still be constructed. These alter your ship's offense and defense in effect, and do who knows what else. I've never actually used any though I have unlocked one or two.

And as of now I have 3 power-ups, two of which Shykin mentioned in the comments:

1. Accuracy - next two shots hit
2. Dodge - next two enemy shots miss
3. Boost - speeds up the intercepter while in pursuit (5 more seconds to shoot down the UFO before it escapes).

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.