Looks like Prostrate spurge.....

It's a little too upright and tightly formed to be Euphorbia maculata. Compare to Phyllanthus. If that is what it is, it makes seed balls on the back of each leaf (though these plants don't look mature enough to do that yet.) If not pulled, (or killed in some other way, before the seeds drop,) the cycle will continue with more plants next year. Weeds this short are hard to thwart by mowing. They do pull up easily but just in the small area of your pic, there are probably at least 100 plants. How much area do you have with this stuff? The good news is that pulling it this year should mean there's hardy any of it next year.

Looks like part of your lawn might be in too much shade to grow well?

Yes, the part where these are growing, is under a jap maple, and there is a bit too much shade for the bermuda to grow well. Here is another photo from farther away. I have been pulling them, but there is so much that I was hoping for something I can spray or put down, which will get rid of it. The entire lawn is treated for pre/post emergent, but I am wondering if I need to complain to the company who applied the treatments. Should the pre/post emergents have taken care of this??

Last question is way out of my area, I don't do chemicals, but somebody should know.

That would be a lot of pulling. Looks like a good spot for a new mulched area to me, probably wouldn't need a smother layer of paper or cardboard to kill the already struggling grass and Phyllanthus, add Hydrangeas, Azaleas, Hostas, shade entities, but that may not be your style. Taking the lowest limbs off of the tree might be an option if you'd prefer grass. Maybe get a bunch of paver bricks and make a patio area? Not sure what would work in the big pic, just ideas for thought...

Spurge - evil stuff!

If you see 'little balls' on the back of the leaves, it's Phyllanthus.