Do you have to pay the 7 to Reaction Deploy the Predator? I know this wouldn't allow the Predator to use its ability immediately, but would it allow you to get around the drain on resources and give you a 5/5 for exhausting a 2/2?

RRG, p. 12:

"Put into Play

Some card effects may cause a card to be 'put into play.' Such effects move the card directly into play from an out of play state.

> The deployment cost of the card being 'put into play' is not paid."

So no, you do not have to pay the printed cost when using a "put into play" effect.

As you already note, using a "put into play" effect is not considered to be "deploying" the unit - which would involve paying the printed cost. Since you do not "deploy" the Predator, you do not get to use its reaction. (Of course, in the overall situation, it would be too late to trigger it anyway because enemy warlord is already at the planet....)

It didn't occur to me to refer to the RRG. Coming from a 40k tabletop background, I sometimes forget how complete FFG tends to be. That is also one of the things I love about them, they actually give a crap. I guess that is why they are a Game Company, while Games-Workshop is not, despite their name.