After snack: "Here sweetie, take this to the table. Nope - come back - we're not finished....here's a cloth - you may wipe the table" and so on. Then "Now we are going to play a game". Lead them through a few circle games. Then have a story. Take the little boy into your lap - he is probably dying to have an adult's warm and centered aura surrounding him so he can relax into his body. Then :"Ok - what a lovely visit. Time to go home now. W will walk you home" Keep the initial visits to no more than 1 hour. CLEAR BOUNDARIES EVERY STEP OF THE WAY!
And at every step of the way you are right in there to the fore. Your presence, your guidance, your calm structuring of the situation will allow play to happen and allow the children to be in your home. This is a therapeutic situation - and I am afraid it is absolutely normal these days. Many parents, of course, don't witness these problems because when children come over, they are immediately plugged in to a machine of some sort. The children are left to deal with the energy built up in them by the images they view all by themselves. And it does come put - maybe at school the next day, maybe it builds up to a crescendo and a huge out burst. And maybe the children get labeled as having ADHD or some other problem when really, they are just being mishandled and not understood and their developmental needs are not being met.
I strongly suggest that you sit down and have a good think anticipating the future. You need to have absolute boundaries (key word these days, eh?!). If they do not cooperate, they go home. Not a threat - a friendly fact - that's how it is. No wheedling or cajoling. Do say things like "ok - you can go run around back at your house. But here - let's play this game first" (Oh - and I do suggest you select your games with care and that you do not have too high expectations - it could well be that these children have never had a chance to play ring games and simply do not know how to play. Again - a very common scenario in the lives of America's materially rich and spiritually impoverished children). You need to work via imitation and being right in there physically taking charge.
The pictorial language will not work on children like this because they are so out of themselves that they are unable to be receptive to the images. They cannot even imitate. You have to build up the foundations for them from scratch. Hard work - but very rewarding.
Think ahead a few weeks or months - ok, you've cracked it, the children can play together. You need to have rules about what happens when someone brings his gameboy over. Or invites your child to their house to "play" ie watch videos.
You need to be busy so that the "dumping by parents" problem you mentioned is nipped in the bud" "Yes - that will be lovely to have X here - but we are going out at 11 so you can pick him up then - great. .... No - he can't come with us - that doesn't work. Great to have him here now - see you later!" And really do go out so they can't call your bluff. If they pull a no-show, take the child to their house and leave him there. You're going to have to be tough to make this work. Tough and VERY clear.
These are therapeutic situations for the parents, too. Imagine being the parent of a child who is off the wall - you'd be desperate to pawn him off on someone else for a few hours! And if those parents don't have the skills and experience to cope....well, it's not that it's your job to teach them - but if this is going to work, you are the one who is going to have to put most of the work in. That's just how it is. Your goal which you articulated was to enable your own daughter to be able to play with the neighborhood children - and this is the hand you've been dealt. There really aren't any short-cuts.
The parents have boundary issues. That's why they dump their children on you. You can't change them. But you can work on your own boundary issues. And if "politeness" and nervousness are getting in your way, then that's your area to focus on! You have to do it! Isn't parenting great?! What fantastic opportunities it provides for our own self development! This is going to be tough, I am sure - but if you manage to get on top of this one, you'll be able to handle anything life throws at you!