I think that material might be hard on young knees; no doubt there will be roughhousing and crawling around to play games. And I agree the off-gassing will be an issue for a tight interior space. I'm a big fan of FLOR carpet tiles for playrooms and basements, and even kids' bedrooms. Super easy to maintain and if anything becomes irreparably stained, you can easily replace a tile. Very inexpensive, too. Here are some examples:

Apparently your neighbor's irrigation system is working and yours is not :-) Kidding aside, your grass looks like it needs just generally better care ... watering, edging, dethatching, etc.... and eventually, more light as the tree grows. That is so cool to have the barcodes on the neighbor's edge restraint. Will be a great way to help future archeologists in your area research people of our time. :-) Meanwhile, I'd slap a brick/paver mowing strip along that edge, matching its height, and then feather the grade on your side, by adding a little topsoil, to the top of the mowing strip. The grass will easily grow through it to the new height. When you do the drive, widen/flare it enough at the street to meet where the curb becomes full height. I don't have an objection to stamped concrete as long as the installer is well qualified. (Too many novices have to cut their teeth somewhere and need to be avoided.) Example. "I tend to like the look of light driveways with darker borders as well." That's going to make the prettier driveways harder to recognize! :-) Having more experience with pavers, I can't speak knowledgeably of fading issues of stamped concrete. Maybe someone who has more experience with it will know and can chime in. I once did a commercial patio where they were trying to cut the budget. We ended up with UNCOLORED stamped concrete. It was surrounded by a 12" wide plain border and the field was stamped with basketweave brick. It was quite handsome and vastly dressier than plain, unadorned concrete. A side benefit was that there would never be issues with chipping either (which if you've seen stamped concrete that has had a fair amount of chipping or surface wear, it can bring the look down.)

Tatts, thank you for reminding me it will be a close-up look. That could be a turn-off. I think that since I have time what I should do is go out to Home Depot (and anywhere else I can think of that carries some of it) and spend time looking at and feeling it. I will probably start out using the rug I linked above. I already have it and don't have room in the new place for it so this seems good. I don't know how long it will last given that it is not an outdoor rug but since I already have it and got it at a fantastic price ... well, why not.

My father has a concrete double driveway that he installed in 1966 and it has no cracks. Niagara Peninsula so it has all 4 seasons. A hard surface as a play area has its advantages, depending on your climate. You can ride a bicycle/tricycle/rollerblades on it. It doesn't get muddy after a rain. But I wouldn't set up a swing set on it. And it can be very hot, unless it's well shaded by trees. It also depends on how you are landscaping the rest of the yard. If there is attractive landscaping and grass to play on elsewhere, and the concrete is attractively coloured then it would just be a large patio for play. If you give us the rest of the plan you'll get more useful advice.