Once the royal icing dries on my cookies, how long will they stay fresh? I'm assuming they don't need to be refrigerated, right?

Especially if you use meringue/egg white powder, you should definitely be good.

Assuming you iced a pretty standard sugar cookie, you should be good for three to five days. Any cookies that are not going to be eaten with hours of cooling/frosting, should be stored in an airtight tin or zip bags. You should also be able to freeze these for later consumption if you like.