Causing the effect to be temporary would be very interesting..  The Advent player would clearly try to use the temp-capped ships to inflict harm on the others, forcing a battle whereas the defender would try to avoid losses, knowing that those are still his ships, he just can't control them right now.  Sacrifice those ships and save the remaining ones, or lose some and keep them when they default to your ownership.

Honestly, I really like that idea.

I dunno, a 25% damage bonus, the bonus mitigation from being in culture, being able to deny enemy culture bonuses anywhere, and a guaranteed loyalty drop on the planet due to suppressing local culture generation (in place of yours) is pretty good.

It's not as flashy as the other two, but combine all that and it's going to give your fleet major advantages no matter where you want to fight. The problem with it right now isn't that what it does is bad, it's that it can't do it most of the time because local culture blocks it.

I really don't see stuff like converting ships, defenses, or planets happening. That could get out of hand very quickly.

That is a given.

While I think totally unfeasible, this would be hilariously awesome.

While it would be horrendously OP, it wouldn't be difficult at all to do this.  Cannon shell applies buff to planet, applies buff to ships, ships explore and spread culture, time expires, ships wake up.  Actually, it's a very simple buff chain.

what if the deliverence engine just set the culture in the well instead of trying to spread it? This way, it wouldn't be fighting any culture already there. then I'd throw an tack an allegiance drain for good measure. Balance can be tweaked by the the duration of the effects, the amount of culture and the amount of allegiance lost.

If the DE could "convert" enemy Culture Centers like a chain reaction it would be also very powerful and usefull.

At the moment you need to negate his culture first before you spread your own, there is the first problem in my opinion, this is to weak. i have seen Vasari with only one propaganda who stand against the DE because his culture was spread to some other planets. after i shot 2 other DEs to nearby planets, i could overwhelm him.
So, if you shoot your DE to an enemy planet, your Culture will spread without negating the enemy culture first.
Also the enemy culture centers get converted to spread your culture instead (enemy needs to destroy them with ships or defense then). If your culture reaches another planet with another culture center, it will also be converted slowly. if there is no culture center at the second planet, the converting time becomes slower. if there is no culture center at the third planet then, it will stop. but with each converted culture center the effect starts again.
Because it takes a looong time to loose youre planet because of the allegiance loss, you have enough time to prevent this. But if you dont want do do anything about it, you will have problems.
Also you can sell the culture centers before the advent culture reaches them, so you can stop the effect until you have destroyed the converted centers.
So it would not destroy wour planets from far away like the nova, also it wont give you the ability to jump directly to your planet like the kostura. but it would freak your enemy if he wont hurry to protect his planets against this slow moving danger.
Because you simply need to destroy the center to stop the effect, it wont be so op. if you have no defense or ships at the affected planets...you have a problem

I think that a simple flip of what the planet considers "friendly" and "hostile" should be all that happens. For a minute or 3, the owner and all allies are seen as hostile, and everyone else is seen as friendly. You wouldn't have to fight internal or external culture centers, but you wouldn't be forcing a conversion either, so outside planets would remain unaffected. Allegiance would decrease like normal, since the culture is considered hostile. Logistic structures are unusable by the owner for the duration, and tactical structures view the owner/allies as enemies. Starbases wouldn't be directly affected, since they aren't linked to the planet. I think the user would receive bonuses while the effect was in place, since said player would be on "friendly" culture, but I don't know if that's linked to the planet or the player. If used on a neutral, the only thing I can see being affected is friendly/hostile culture, but that goes back to whether culture is linked to players or planets. 

Unfortunately, you lose the functionality of provided a quick fix if your own culture is getting low. With the above being said, it goes without saying that you shouldn't use it on an yourself or an ally (Unless, of course, you're preparing a backstab )