Unsure and Need Advice

Okay, I am new here, but I am hoping some of you guys can help me figure some stuff out. So I started seeing someone, I think. We met a week ago and have hung out almost every night since. He is, well he is a character, lol. He usually doesn't like people, but we hit it off. He says that he wants me to like him as much as he likes me and that I have absolutely no reason to be apprehensive, and after I told him about a bad experience with someone he told me he would never do what that person did. He told me to let my guard down and have faith in him, so I am trying to do just that, but it is hard. Last night I went over to his place again, after he had a terrible day, we had some drinks and played video games and had tons of fun. He said that it makes him mad that I think he is going to treat me like other guys, and I said I was sorry but just like he keeps saying I am too good to be true, I think he is too. He lets me stay at his house when he isn 't there and cooks for me (which he says he doesn't do for just anyone). He gets aggravated because I am not as trusting as he is with me. Then, a couple of nights ago, he asked why I don't want to fool around with him and I said because I wasn't his girl. Then he was like you don't think you are my girl? I thought it was kind of implied, I mean we haven't had that conversation yet but I figured you knew that you were my girl. Then he was like do you want to be my girl, and I said yes, I have been waiting for him to ask and he said then you're my girl. Then we proceeded, to well, do stuff. I didn't go over the next night, but I did last night as I mentioned above and everything was fine. Then he jokingly said he had to kick me out because he had to leave and he kissed me goodbye and told me to have fun packing my stuff up with my mom (we are moving into a new house). He doesn't text a lot because of his job and he has a pretty busy weekend, but I just have reservations and kind of would like an outside view. First, if he said I am his girl does that mean we are official? And secondly, does it seem like he is really serious?

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