Does 1 Corinthians 7:12-16 imply that if a believer marries a non-believer, that the non-believer can be saved through that marriage?
Yes it does, but we should be careful how we interpret this. Paul is saying to the Christian sister who is married to a non-Christian that she should not leave her marriage and seek a divorce unless her husband sends her away. He tells this rather vulnerable sister that it is possible that if she stay with the man, he might become a Christian. He is NOT saying that her righteous acts could save her husband but that if she stays with him, it may be that he will respond to her convictions and decide to become a Christian on his own. This is not a promise but is simply given as one reason she should stay with her husband. Essentially, he is saying to her to trust God in staying with her husband, as he may work through her to help her husband.