So how come Germany would vote NEVER! for ending the war with me, considering how much Germany likes me? Also considering we haven't even fought once.
I'm honestly not sure on this one. I usually work around the AP timer when fighting wars where it is a factor, or possibly at the point I've taken vassals I already control the AP. Could be a random choice or based on certain factors like his attitude toward the enemy (in other words, he wants the war as much as you) and/or war success/war weariness.
I am creaming my enemy, I can get them to capitulate and become a vessel and surrender one of the two cities they have left, leaving them with the worst city. I'm only a few turns away from capturing the city anyway and another few turns to wipe out that nation altogether. Does it make sense to make a deal and get the city 2-3 turns earlier? It leaves more structures in, right?
What structures remain in a captured city is random, so I don't see waiting having much bearing on the matter. Taking on a vassal with just one city does not seem worth it other than just to end the war and move on. You can gift the vassal back some of his cities though. Culture pressure can become an issue though, possibly. At the point you are in in your war I'd probably just finish the guy off.
Is there a benefit for having a vessel?
They are very nice for ferrying troops over water
If you mean vassal, there are some benefits to vassals as follows:
1) Depending on the state you leave the vassal in at capitulation, they can tech stuff for you. In the dialogue you can actually ask them to tech what you want. Leaving them with a decent number of cities helps, so as mentioned above you might gift them back some of their cities to make them functional. Gift them some economic techs they may not have as well to boost them.
2) Vassals can be meat shields in war and sometimes do some of the work for you - though this can be a bit unreliable and not easy to control. Depends a bit on the leader too and how aggressive they are in building units. In the dialogue you can ask them to attack a particular city - again can be unreliable - but they will usual send some units, while at minimum they can be a distraction for the enemy and make your war goals easier. Make sure you advance them militarily with tech and any extra resources you have that they don't like iron or horses.
In some cases, I've even gifted my vassals units especially older units so they can spend the gold to upgrade them. Though I don't necessarily recommend it, a funny approach is to cut off or trade away your metals and gift your vassal a bunch of warriors..they will upgrade. (more effective on high levels where AIs have more gold and upgrade costs are less)
3) Each vassal provides +1
to your empire.
4) Vassals count toward conquest and domination. For conquest, a vassal is a defeated foe. You may destroy some civs or vassal some civs but when all other leaders are your vassal (or dead) then you've won by conquest. Taking vassals can speed up conquest victory since you cap a leader asap and move to the next, as opposed to wasting time taking all their cities. For domination, a vassal's land counts 50% toward the land requirement for domination victory. So again, can be a quicker way to domination as long as you are aware that you need to make up the difference in captured or settled land.
...related to the above is another type of vassal...the colony. Colonies can be created from cities on overseas landmass or islands. Colonies are basically vassals that start extremely loyal and happy with you when you liberate them, and get all the tech you have at the point you release them. One benefit of overseas Colonies is basically claiming land toward domination without paying all the overseas maintenance costs.
5) Vassals always vote for you for AP/UN leader/DV resolutions, unless the off chance they are a rival for that vote, which can happen.
We use the term Diplomation to apply when when uses war as a means to secure Diplo Victory by way of vassaling most everyone.