Stations and ships. Is there any way to get them to coordinate together? It seems every time I have any ships on the same tile as a space station and someone attacks that tile they only attack ether the station by itself or the ships by themselves. Never both at the same time. I have yet to attack a tile containing both a station and a ship.
Fleets and ships take attacks before a starbase does. Unfortunately, there is no way for a starbase and a ship or fleet to work together.
Trade goods. If you build a trade good is the effect empire wide or does it only effect the planet it is built on?
Yes, trade goods provide empire-wide bonuses. There are some other unique improvments that are planet-only, e.g. Hyperion Shipyards.
Repairing damaged ships. I know you can accelerate the repair amount by parking them in orbit around a planet, but does the repair amount increase if that planet has a starport?
Not that I know of, but I'd love to be corrected if I'm wrong.
On the two construction or two trade modules question, no to both. You waste money and space if you add more than one colony, constructor, or trade module to a ship.
Faster trade ships will get a route established more quickly, but after a route is there, a 'mini-freighter' replaces the unit you built and always moves at speed 1. (If you add non-engine features to a freighter like sensors, weapons, or defenses, the minifreighter *will* have those abilities.)
Are there any benefits to forming an alliance with another race if you're not going for cultural or diplomatic victories?
Probably depends on your play style, but even if you're going for a research or conquest win, you might sometimes find it handy to have an ally who's stronger than you or a weakling ally who might give you a 'legitimate' excuse to start a war you want without seeming like you are the aggressor.
What do economic and research treaties do?
Basically two things: they help you earn more BC or do more research, and they improve your relationship with another civ. The BC and research bonus amounts can vary widely depending on how large the civs are, but even though I prefer the largest maps, I mostly think about them in terms of the diplo context and count any bonus resources as gravy.