1. What game mode are you in? That makes a bit of a difference. The last stuff that is unlocked is I believe to do with the RCI bars. Basically the tutorial shows you how to play mechanically, but doesn't guide you through strategy too much. Which might have been an oversight, but it's hard to give too much guidance of that other form in a sandbox. Basically: make friends with some folks, get them into the federation, and then backdoor the others into the federation or smash them. The Skylaxians are excellent at backdooring others in.
2. Yes. If you kill the prime warlord, then one of them will become the new prime. Sometimes it can be easier to get leverage over the secondaries, so you can always turn one of them, then kill his boss, then twist the screws on him.
3. It's in the political deals, and it's something that usually requires moons to be colonized first. Not always -- but moons are good sources of resource income for the races in terms of specific types of resources, which is what they need in order to trade. Exactly how you set up trade deals depends on the race you're talking to. Only certain CEOs will discuss it, for instance, and only certain Andor parties. But the Skylaxians and Peltians and some of the others always will. So sometimes you'll want to go to them to start something up. Anyway, it's in the political screens.
5. The fast forward button is only present when you are in a contract of some sort. If you're not in a contract of some sort and you fast forward, you're wasting opportunity! At least mine something or improve somebodies RCI or what have you. So you go in there, and then immediately you can FF as much as you like.