How do leaders gain experience

As far as I know, you can't, you have to watch the movements of the foreign fleet. But here you have two options:

1. 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.

2. 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.

I had a similar problem. I vassalized my rowdy neighbor and integrated him doubling the size of my empire. Only problem was they were a bunch of xenophobic slaving ****s. Try this;

1. Click on the rowdy planet and go to the planet summary screen
2. Check out the edicts list.
3. Happiness and Ethics Divergence.

Happiness will decrease the likelihood that pops join a faction. Ethics Divergence will change the traits of their pops over time. Usually towards your main race's traits.

Give it a few years. Keep pumping up happiness. Change government types if it increases happiness or ethics divergence. If all else fails, station an army there and wait for the revolts. When you crush them, they'll shut up for a while.

Hope this helps!