Suppose figures A and B can merge. Figure C can mind contol friendlies using the Compel feat. If A and B merge they still have no action tokens. Can figure C compel them to split on the same turn? Ordinarily I know they couldn't, but I hear people talk about how mind control "resets" powers, which I've never fully understood. Would this be one of those cases? Could they split on the same turn they merged using friendly mind control? Would they take action tokens?

Sorry not to quote powers, but I'm on my iPod.

From ye olde PAC:

So, then why do once per game actions reset when a character is mind controlled?

A character can be mind controlled to Split/Merge but the player who controlled the figures originally still gets to choose who those figure Split/Merge into. It used to be the choice of the controller of the figure that did the Mind Controlling this became a huge problem. because players would pick the worst two possible figures for the Split/Merg royally screwing their opponent to the point where matches became near auto-wins.

Because of this line in the Player's Guide:

When a character is affected by Mind Control, any of its
powers previously countered remain countered. Powers
and abilities that may only be used once per turn, round, or
game may be used accordingly per Mind Controller.

Split/Merge isn't a "once per" power or ability. Using Merge simply prevents the character from using Split on the same turn. These issues are not related at all.

I've only used Merge/Split one time before so I've never needed to ask this... after you Split a duo, can they be given a free action or no actions at all that turn? I assume the latter based on the wording, but I wasn't sure if there was a clarification that I've missed.

SPLIT
Give this character a power action if it has no action tokens. Replace this character
with one or more qualifying characters each from a different entry listed on
this character’s character card. The qualifying replacement character(s) must
have either qualifying name and total no more than this character’s point value
(unless those characters previously used the Merge ability to be replaced by this
character). Replacement character(s) are assigned an action token and can’t be
given an action this turn. Replacement character(s) begin a number of clicks from
their starting line equal to the number of clicks from this character’s starting line. This ability can’t be countered.

Action = any action at all, of any type.

If it were to still allow free actions, it would say "can't be given a non-free action".

That's what I assumed, but I wanted to make sure I didn't miss any clarification.
So you couldn't friendly MC a Split figure, but your opponent could MC them.