Well in order -
1) If I lay mines, will they explode with the ships of my partner (allies in partnership)?
No, mines will not explode if you have any positive treaty with the other empire (i.e. they only explode if 'no treaty','non intercourse' or 'war')
2) What is the use of satellites around planets compared to weapon installations?
The number is only limited by the number of units allowed in the sector (forget the default value but quite large!). Not by the space on the planet (as long as you launch them!). Though WP need to be destroyed before any damage is done to the planet unlike sats, and WP are generally better if you don't have a space problem.
3) I have researched troops. Do I need some specific ship component to make planet invasion?
Just put them on a ship as cargo. Them get that ship next to the planet (in tactical combat, for strategic it depends on the stratagies you set for the ai) and give that ship the order 'drop troops'. Note you can drop troops on a planet with WPs, they only try to shoot you transport down!
