How do I check another empire's FTL method?
Solution 1:
As far as I know, you can't, you have to watch the movements of the foreign fleet. But here you have two options:
If you yourself use hyperspace, you can see the hyperspace lanes, if you don't, you can research a technology (hyperspace mapping) allowing you to do so, though it's a rare tech. Either way, once you can see the lanes, you can easily determine whether an empire uses hyperspace or warp by seeing whether they travel on the lanes or not. Or wormhole, if they don't travel at all, but simply appear.
Another factor is the cooldown time. Hyperspace and wormhole have a charge-up time before the jump, whereas warp has a cooldown after a jump. When a foreign fleet enters your system and you read something like "warp winddown" in their status window, they use warp. If they are able to move immediately, they use hyperspace or wormhole. But you can easily distinguish hyperspace from wormhole by the presence of wormhole stations and the fact, that wormhole fleets appear instantly in your system instead of travelling through space.
Solution 2:
When you click an enemy fleet, it has an information, then you click on one of the ship to view ship design. There is one slot of FTL engine that allows using it.
Bonus:
How to counter (this assumes that you have free transportation)
Warp Drive: Use space station (combat one) that bring enemy into the station then kill it quickly or at least make them less powerful before warp wind down.
If you're their friend, they won't have much trouble traveling except if they're surrounded.
Hyperspace: If you use hyperspace mapping, you may block their way if you're their foes, and heavily fortify it if you're their friend, then it would be a pleasure to allow their access or leave this system theirs.
Wormhole: Split your army and no matter where they came from if you can destroy their wormhole station they're stranded. If you're your friend station right might be good or they may never come to help protecting you. Also this wormhole make you can go pass any territory into system with station. If you're wormhole player you might build a strong economy by building everywhere with rich resources and still get connected.