What kind of box should I put these in?

Unless you get boxes made for cupcakes you're going to have problems keeping the icing on them from mucking up. I make gift baskets for family/friends every year and for that purpose I never make cupcakes because of the icing. I do give cupcakes as gifts on other occasions but I use boxes. You can in fact get boxes for 4 or 6 or 12 cupcakes. MIchael's may not have them but depending on where you are located, other similar stores will have them. Costco, Bulk Barn here in CAnada has all sorts of boxes/storage options. I don't know where you live so I couldn't possibly suggest places to shop. But there are cake decorating stores everywhere you just have to research this.

For my gift baskets, I use actual baskets, they're a dime a dozen at any dollar store. I line the baskets with pretty cloth or paper, and then I organize the goodies in the basket. I always keep a big roll of the clear gift wrap and ribbon. And voila. My basket sizes vary because each family is different size, so I allow for a couple of truffles, cookies, etc, each person.

I would pack the cupcakes alone, then find a separate container for the cookies, so that neither gets squished.

Cookies take on the flavors of their neighbors in a container really quick, so slip each kind of cookie into a plastic bag, tie with a ribbon, then add them to the container.

Go to your local Dollar Tree or Dollar General- they have plastic cases that open and close specifically for cupakes, usually in sets of either 4 or 6.

skip the cupcakes, they don't pack and stay too well. Brownies and any other bars and cookies package much better.

if you're in uk, then poundland are selling round servers with a lid for cupcakes/muffins.