Both "go to great pains" and "take great pains" (and their conjugations) are valid. I'd say the latter is more common, but Merriam-Webster attests to both forms. "took [noun] great pains" seems to be a possible but uncommon use.
I'd say definitely. (Others appear to disagree.)
As regards the "go to" option, it appears that people do say Go to great pains -- but I wouldn't.
I might say
He went to great lengths...
He went to a lot of trouble...