I don't think it's silly and childish to worry about this sort of thing. Lots of people have relationship anxieties, even adults; it's perfectly normal, so don't worry about that. I can see why it'd be especially worrisome if there wasn't clear communication in your last relationship and you only found out at the very end that your partner was unhappy. That's bound to cause insecurities and I can see why you'd want things to be clear with your current partner.

It sounds like there's good indicators that this is something more than a friendship if you've kissed and held hands. However, I do want to point out that it's still really early on; it's only been a week. Personally I feel it may be a bit too early to have the "are we boyfriend and girlfriend" talk if he only just told you he likes you last week; he may feel a bit of pressure if you ask that right now. It sounds like to me you're having a good time together and that's what matters most. While it might be frustrating to "not know" what's going on for the moment I would give it a couple of weeks before you ask him if he's your boyfriend. Give things some time to settle and that may be a more successful conversation than it may have been otherwise.

I do, however, think that it's perfectly acceptable to ask about displays of affection at school, especially since you're so anxious about it. Good relationships are built on honest and open communication. It's a process that happens over time but it's good to try and make that a habit now. I suggest approaching it casually, perhaps something along the lines of the following:

"Hey, I just want you to know I've really enjoyed spending time with you this past week. I'm really happy with how things are going. I wanted you to know I'm having a bit of anxiety, though. I know at this point only our friends really know we've gotten as close as we have. I was wondering how you want to act around each other at school. Is physical affection okay, or would you prefer not to engage in that sort of thing at this point?"

I think something like that would open up a dialogue were you could work together to find something that works for you. It's a reasonable question and I hope he's open to hearing what you have to say.

Good luck and PM me if you need anything else.