So are missiles a bad idea even if the enemy has a missile weakness because they take too long to get to the target?
All weapons work instantaneously in GalCiv 2. The only time a destroyed ship doesn't return fire is, if one side has the Super Warrior ability and initiated the battle. However, that only works on the first turn of combat.
Is there a way to try things out without having to build ships and go into battle?
Well, there is a combat simulator in the Game Talk sub-forum. However, it was only written for the base game, and doesn't include the changes made to combat in the expansions. There is also a post by one of the devs detailing how combat works in the expansions. That will probably be of more use to you.
Is that why I needed a vote to go to war? Or is a vote always needed?
You only need to vote to go to war, if your government type is not Imperial (i.e., if it is a republic, democracy, or federation).
What's the best way to provoke a war so that you don't loose reputation?
Place some war-ships (preferably with troop transports) around their planets. It may take a while before they declare war, however, depending on their current relations with you, and on how strong their military is compared to yours.