TEC
Interceptor:A new crusier that has "strikecraft" that lockdown trade ships and
refinery ships and steals the credits and resources. Seems logical
that the TEC would have a unit that further increases the economic
advantage they have.
This ability ALREADY exists - it's the Sova Carrier's Embargo ability!
Vasari
Enslaver: A new support frigate that can use an ability to steal resource asteroids from its enemies. Bascially sending in a crew to enslave the current workers. This would work on both neutral gravity wells and hostile ones. forcing the prevous owner to blow up the current processing plant and build a new one.
This ability already exists with the TEC Colony Frigate and Vasari Scout when dealing with neutral mines. I think it would be better if there was simply a deep research upgrade that allowed the Vasari Scouts to enslave hostile mines.
Advent
Gospel Frigate: A new crusier or frigate unlocked deeper in the culture tree that "bombs" a planet with Advent religion and knowledge - what it does is drains health points from the planet without killing any population. This seemed like a logical ship because the Advent are out to convert the masses.
I like this idea a lot. The benefit here would be that once the planet is converted, you wouldn't have to wait for the population to increase, nor would you have to spend money to re-develop the colony.
A later upgrade for this ship could be the ability to go immobile and act as a broadcast center.
Maybe instead if they had a Suicide Run ability that allowed the Gospel ship to sacrifice itself by smashing into another ship might be more in tune with the Advent's religious zealotry.

As for the other ideas, they're too similar to abilities already offered by other ships, and as such, don't really warrant a whole new ship.
Personally, I'd like to see the incorporation of espionage into the game, which would allow you to do covert missions against enemy and even friendly factions. For example, you could sneak into enemy territory and blow up one of their structures or disable one of their capital ships for a while, perhaps even while posing as another faction. The ability to capture enemy ships and use them in these covert endeavours would also be awesome. This may not warrant new ships, however, as these abilities could be wrapped into the existing ships under research upgrades.
Mad Cat