1) If a cohen marries a shvuya are the children chalalim? 2) Can one be mevatel issur lechatchila through a non-jew?
1) The children are safek chalalim; boys are forbidden from marrying divorcees, and girls are prohibited from marrying Kohanim (Even Ha-Ezer 7:1, 7:20). Of course, the ruling in specific cases depends on many possible permutations and circumstances.
2) It is forbidden to annul an issur lechatchilah by means of a non-Jew, and one who does so is ‘fiined’ in that the mixture remains prohibited (Birchei Yosef, Shiyurei Berachah 99:6, in the name of Radvaz). However, if a non-Jew annuls the issur without the Jew’s bidding, the mixture is permitted.