Wow, I would hate to live in a neighborhood like that. We don't live close to each other (out in the country) but we don't care if a kid runs across the yard either or if a bunch are riding bikes on our long gravel driveway.
I am always interested in these different neighborhood cultures. Why does anyone care if a neighbor kid is on their grass? (not meaning the OP's neighbors. If a kid is causing trouble I could understand. But just a general kid)
I'd hate it, too, and thankfully our neighborhood isn't like that for the most part.
As far as grass, some people are obsessed about it! One of our neighbors is like this and he has 3 kids. They are very particular about people on their grass; they want it to stay lush and green, no patches caused by kids running repeatedly across the lawn. Me? I don't water our lawn, we've got weeds and bald spots, but my kids can play freely. I think that's more important that having a picture perfect lawn. I don't like when the neighbor kids or my own trample my flowers and generally speaking they are careful, but I don't restrict them from my yard because of it.