Move out. I ain't kiddin'. This is early to be living together. Have some independence from him. He just seems to be rushing into things, particularly considering the situation with his ex. You don't want to be picking out curtains with him thinking to himself, ___ would have picked the blue ones.
He might miss you when you move out. He might not. In the meantime, date him but also have a conversation. He might not be able to articulate where he is emotionally, but you need to know if he is committed to you. If he is not, then I would also suggest dropping exclusivity (on both sides) and date others. Give yourself options. I know you want to skip ahead to marriage but he does not sound as into it as you are.
Yes, he can look up other people online (I do this all the time and so does my husband), but for him it doesn't seem to be harmless curiosity.
Move out, also, before you commingle your finances and possessions any further.