The first weed is some kind of Oxalis. The 2nd one is some kind of Tradescantia I think. It grows where lots of water saturates the soil, grows on stilt roots if it has to. Maybe you are watering too much?

So far as using copious amounts of a weed and feed, if they kill your weeds dead, what else will they affect? The tree may be being affected, but maybe not.

St. Augustine grass is cheap, easy to grow and is installed because it makes instant lawn. A very inferior grass for a lawn, it makes a much better vine. Really.

Because of the large areas of lawn with bald spots (St. Aug. grass lawn is mostly blank space), the best advice I have is to feed the lawn and provide good drainage. After awhile, the weeds will retreat. I know it is the practice in FL to constantly obsess after a thick lush lawn, but it comes at a high maintenance cost. When I lived down there, I fed the lawn weakly about every other month, and never used any weed killers. In the cool months, (Nov-Apr) I mowed low, in the hot months, I mowed high. After 2-3 years, people were asking me how it looked so good, because they were doing the weed and feed cycle and still had a lawn with bare spots, weeds, and were spending money they didn't have for lawn services.

Agree #1 is Oxalis, possibly 2 types, some of which are natives. (The one flower doesn't look like what I would expect from those leaves.) That purple coloring looks like a health issue, nutrient deficiency or effects from the chemicals.

#2 if Tradescantia it would make a 3-petaled flower, likely pink or white. Also looks like Commelina communis, which would make a blue flower with 2 petals. Do you remember seeing any flowers on these?

The growths on the tree are just new branches. Unless you want new limbs there, you can cut them as close to the trunk as possible without damaging the bark.

It looks like it may be time to sharpen your mower blades.

Yes, Commelina. I couldn't think of that, but I suspect it is Asian Dayflower. When I lived in FL, this weed grew in the gutters of some houses, lawns, cracks in the concrete. That and Moses-in-a-boat (Tradescantia species).