Should i watch out?

My boyfriend started talking to this new girl. Now I don’t care who he talks to. But we did set some things to follow in the beginning of our relationship (things he asked me of, and i agreed as long as it went both ways). He wanted me to let any guy i talked to know that I’m in a relationship. And I do that anyway, he didn’t have to ask me. But i expect that as well. Well my boyfriend has a tendency to not do it. And i asked him about it. He said she had asked him if he lived alone in his apartment and he said no i live with my girlfriend. Well today, they’re talking and talking about dogs because he wants his own. And she asked if i had lived with him because she was just curious of who’s going to take care of him while he works. So that confused me a bit. I brushed it off and figured she might’ve just forgot. Btw I was sitting next to him so thats the only reason I know what they were talking about. I do look through his phone occasionally and he does mine. We both said we could. It works for us. BUT he always keeps his conversations (on snapchat) for 24hrs. Just to reassure me, is what he said. Then he took off the 24hr chat. They talked for a bit and he put it back on. Later on he said he did it to see if i trust him, but I find that very odd. He doesn’t do that. I’m very curious as to what happened during that time because he walked into the bathroom and just stood there like 3 times and just stayed away from me until he put them back on. I’m a little worried, but not too worried as she knows he is with me, but still a little because i know some women don’t care and do what they want anyway. Should I just kind of keep an eye out for anything else a little off putting? Should I go ahead and ask him about it?

I feel like if I ask him, guilty or not, he will get mad. So i kind of want to just keep an eye out to save an argument if nothings actually going on.