The question: My front lawn, which contains a honey locust tree, is beginning to sprout all sorts of little plants since we tried not to mow during the heat wave. What are they?
The answer: You did the right thing by halting lawn mowing when it's incredibly hot. But if those sprouts have a nice lacey-frondy look, they are probably from the honey locust, an incredibly fast-growing tree much beloved by civic park departments because of this very trait.
Honey locusts are not long-lived and so have to reproduce any way they can, including suckering madly about in nicely watered lawns and sending out massive numbers of seeds (much loved by bunnies and deer). My advice would be to get rid of the tree and get something more sensible in there.
The Gingko biloba, for instance, is well-adapted to dealing with pollution, has deep roots and will put up with neglect once established.