When you give a character a double power action, and that character does not have willpower or indomitable, Will he receive 1 click of push damage

Yes they will. When a character takes that second token, you make a check for Willpower or anything that gives it. If you have WP, no click; If you don't, then you take it.

Yes, he will. There's no reason he wouldn't.

Now does a double power action count as 2 actions? Say in a 300 pt game, if I do a DPA, will I only have 1 action remaining or does it only count as one?

It only counts as one.

There are a couple figures (note, only a couple) that have powers that say you give the figure a double power action that does no pushing damage so there are some options.

As a side note, you can ONLY give a DPA when a power offers it... you can't just give any old figure a DPA to take a push. I wish you could as it would make those activation click figures MUCH easier to deal with... but you can't. :-s