I tried searching for this, but I didn't get anything, so I thought it might be worthwhile to ask.

If Tiamat uses TK to throw Headpool using Headpool's movement special "Hail Mary, Boss!" does Tiamat lose a token and take a free action if he hits per his "Champion and Steed" trait? Or is Headpool the one doing the damage and will be placing a Virus token?

Related, since he was used in a ranged combat attack like a light object, does Headpool still have the option of taking an action after he was thrown like if he were placed, or will he be taking a turn off?

Tiamat is using TK to make a ranged attack (with Headpool). If the Tiamat's attack hits and the target takes damage, Tiamat can be given the second action.

Headpool can be given an action after the TK action resolves.

Headpool can still be given free actions and non-free actions that turn after he has been placed by Hail Mary, Boss.

If damage was taken by the target, Headpool would not hand out a virus token and Tiamat would lose an action/be allowed to take a free action.