1. Ctrl-F3 when you start each session, then just leave it open. Especially because an event happens every time advance - you probably mean 'an event I consider important enough to be notified about' and figuring out what that entails is quite difficult. Although you can turn on or off notifications about hundreds of categories of events, it still isn't split up the way I would want it.
Anyway, you can also get a list of the current time advance's events to show on the F3 'System' screen, if you tick the option box on the display tab. Of course, taht only helps if you're the kind of person who plays Aurora from the F3 'System' screen.
As an aside, Steve plays on a dual monitor setup, with the events list always open on one side. I play with the F2 'Population & Production' open on top and aligned to the top of the screen, which leaves just enough room at the bottom to display 5-6 lines of the Ctrl-F3 'Events' window.
2. Yes, and yes. Keep in mind that anyone can use anyone's Jump Gates, so building them is also known as 'building a highway for invasion'.
3. Armour 4, commercial engine, 1-2 geo survey sensor, 1-2 grav survey sensor, fuel tanks, engineering, commercial jump engine big enough to move the ship (Note that the grav sensor makes it a military design, so it will need to be built in a military shipyard) - then adjust to taste. If your ship is too slow, add engines. If it takes forever to survey a location/planet, add more sensors. If it runs out of fuel halfway through the second system, add fuel tanks. If it keeps breaking down and exploding, add engineering spaces and maintenance storage bays. If you encounter a thick nebula, add armour to keep the speed up. If you add anything, you'll probably need to increase the jump engine, unless you overbuilt it in the first place.
If you want it to be good at First Contact situations, add powerful (large) Active, Thermal, and/or EM sensors. Don't bother with weapons (though you can include CIWS if you really want to) because you can't mount enough of them to be useful. Though I suppose if your empire is genocidal enough a planetary bombardment weapon of some sort (10cm Meson or Sz 1 missile launcher) can be useful 'pour encourager les habitants!'
4. Geo teams only work on Colonies, so make sure you've added one to the asteroid in question. (Colonies, not populations - there is a difference.) Double-check that they didn't finish the survey years ago and you just missed the message - or if you aren't sure, move them to another colony.
Max team skill is 1275 (255 * 5 team members).
Personally I consider 'training up' Geo teams on less-valuable rocks to be 'cheating', but since the chance of exhausting a system body is illogically based on the team's skill, I can see the argument the other way.