Around here, a play date for that age is a drop-off, unless the invitation is for a child with whom you are not familiar. In that case, one mom usually invites the other over for coffee and the kids can play. Once you are acquainted, I can't imagine why the other parent would stay - the whole point of these dates is so that one mom can get something done! Of course, if the moms are really good friends and WANT to hang out together, that can be arranged, but you don't sit in the same room with the kids!
Yes, you feed them. You ask the other mom if there are any food restrictions (that covers allergies and gives you a sense of what the other kid eats). If the weather is good and they can run around outside, or if you have a playroom or basement area, I don't see why you need to structure activities. Kids don't PLAY much anymore - there's too much structure and there's no creativity. You can make some things available (art supplies, for example) but then leave them alone.
For 7 year olds, a couple of hours is fine - 2, maybe 3.
The rules of the house are the rules - your kid follows them at another house, the guest follows them at your house. You need to let the kids know the parameters - only play in the backyard, stay out of the garage, all snacks and painting are done in the kitchen, etc. For 7 year olds, I don't think you need to specify no hitting, no swearing, etc. - just intervene if it occurs and say that we don't allow that in our house. They know this stuff from school.
I don't think you need to be in the same room - just within earshot. If there is a swimming pool, obviously an adult needs to be present for safety reasons. They don't need to be watched every second, and play dates are to encourage socialization among peers.