It sounds like he’s distracted with family issues. If he won’t talk to you about them, then you should understand that he doesn’t feel close enough to you to divulge them. This could be temporary because he doesn’t feel the two of you are close enough to do that yet — or it could be permanent and this is just who he is when it comes to family. Or, he may not be as into you as you imagined and this is the way you’re figuring that out.
Usually when a couple is moving towards something serious they talk about their families because they want to share and they want you to be part of their family and understand the dynamics. Since you’ve been dating for a year now, and this has come up, it’s a great opportunity for you to consider if this is someone you want to continue dating or not. You don’t have to feel trapped because you’re not married, you’re just dating. But you should see this crossroads as an opportunity to ask, learn and reconsider.
As for your questions about this other woman, you’ve asked him about her and he’s answered you. I think you’re right that it’s important not to pick at him or your relationship with him because you don’t completely believe him — because he’s given you his answer. For now, you have some observing and considering to do! Hang back, ask and listen, and stay open to all possibilities. I hope that helps.