Getting rid of grassy weeds in grass is a problem. Every now and then there's an herbicide that targets one grass over another, but not often. Then it comes down to tough love. If you use a spray on herbicide, you'll have to kill at least a little of the surrounding grass to kill the weed. If you wanted to kill it and then reseed, now is the right season to do that.

I'm certainly no expert, but I know you can get a clue if you look at the shape of the stem between where the roots begin up to where the leaves branch out. If it feels triangular or some other shape that helps to narrow it down. You can find information for weed ID on sites like: http://weeds.cropsci.illinois.edu/weedid.htm I think it is a slow day in the forum. I'm sure you'll get some good replies tomorrow.

Yes, that looks like it and would make a lot of sense, so it's almost definitely that. From what I understand this should be ok considering what the owner wants. I've already offered to enhance the turf, and even though I'm paid well, he simply doesn't want to at this point. Japanese Stilt Grass sounds like it should be able to handle mowing

Could be "Mallow neglecta" a fairly common lawn "weed" sometimes misidentified as ground ivy. The "weed killers" probably killed off the plants that were there but new ones appeared from the prolific seeds these things produce. Perhaps this article might be of some help. http://www.better-lawn-care.com/common-mallow.html#axzz4mEYDCWro