Can an entity possess a duo figure? I am guess no since possession requires the possessed to have the fist or sharpshooter symbol.

You got it right.

There is a first time for everything.

Follow up. If I possess Iron Man (for instance) and merge him with War Machine, is the duo figure now possessed, or does the possession simply end?

My guess is that it would not be allowed to merg because it would create an illegal situation.

There is at least one duo in the game you can possess. That is #046 Batman and Green Arrow duo from the Brave and the Bold set, because they have the -symbol in addition to the -symbol.

Correct. You can also possess the Trinity of Sin Team Base for the same reason.

And if possessed, if you split or do a solo mission, are the split figures/solo figures yours or your opponent's?

Huh? I'm not sure what you are talking about.... Are you referencing the old Possession traits of Deadman and Jericho? If so then that is not what this topic is about

Also, they changed the rules. If you take control of an opponent's character you cannot force it to use Solo Mission or Split/Merge.

Thats what I was talking about Dragon, Thanks!

Now that it has been established that there's at least one team base and one duo figure that can be possessed by Entity, how would that work with Solo Adventure or Split? My presumption would be that the possession would act like when a duo figure has a relic and does Split.

Nothing would happen Solo Adventure. There are very specific rules for Split/Merge on how handle things that were assigned. Those do not exist for Solo Adventure, so the guy who splits off gets nothing.

For Split/Merge treat it just like a relic/resource/other. If you use Split you choose one of the characters and that one character is now the possessed character.

"You get NOTHING, sir! Good DAY, sir!"

"You're a cheat, and a swindler! How could you do a thing like this?"

I could see a case being made for The Trinity of Sin as their ability says they have the SYMBOL, but not for Batman and Green Arrow. There ability says "Batman and Green Arrow possess the Sharpshooter ABILITY" which going by the PAC is described as this;

If the Entities are anything like the Vehicles where they check for specific combat symbols, then Batman and Arrow are a no-go.