Okay, I've read over both questions a few times and I think I know the answers:
The first question would be "Taking" because everything else in that sentence points to a past tense except for the word "taking" which is a present tense thus meaning its inconsistent with the sentence.
The second question would be "After we moved to montana" because its why they had to build the log cabin.
I hope my answers are correct and I wish you luck on your test!