Does the type of kill matter for promotion

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.

I just reloaded my game about 10 times trying to figure this out and I have some results. I tested the same 5 colonels, randomly mixing them together in testing, every single one of them was psi gifted, but not every time. It is not determined when the character is created, the fact that the colonel with the lowest will became gifted but only once out of the 10 times points to randomly gifting the soldiers being tested but adding their will score to the roll.

Simply put, if you want your favorite characters to all be psychics it is easy, it will just take a long time, you will have to put them through testing by themselves or with rookies added in and keep reloading until they become psychic then work on the next character. I have gotten 2 out of the 3 as gifted in about half of the 10 tests, but I was not able to garner a 3 for 3.