Second question first: You'll want a 2" receiver. Reputable shops will install either a 1 1/4" or 2" receiver depending on the trailer towing capacity of your van. If you have a full size, rear wheel drive van, I don't think that will be an issue.
First question: I think that Rhode Gear used to make a receiver rack to carry 5 bicycles. Here's the downside: It's easy to load the first bicycle onto a rack. The second bike is a little harder because you have to be aware of the crank and handlebar position of of the first bike. Size and frame configuration are additional issues. The third bike gets a little more difficult to load yet and so on up to five.
Have you considered a trailer? I'm thinking the cost would be similar to the combined cost of a roof and receiver rack. With a little bit of ingenuity you could securely load all seven of your bikes at a convenient height. You will also retain easy access to the rear hatch on your van to make it easier to get at your helmets, water bottles, pumps and such. That's a HUGH benefit.