I'd keep the maple, daylilies, the Sedum and the daffodils behind the Sedum in your last pic. There's another little Sedum behind the hose. Directly in front of the big Sedum is either a Rudbeckia or Echinacea, some would save that one and since there is only one in all your pics you might want to save that one, I wouldn't though. I noticed the dog strangling vine in one picture but there is also vinca growing under everything in that picture as well, you'll probably have to dig that out because it doesn't respond very well to Roundup. The dog strangling vine doesn't either. I think if I was just getting started gardening here, I'd leave this spot for last and just weed whack it for now, lol. Unless it's in a prominent spot. It looks like the gardener before you took most of the good stuff, luckily the beds look fertile, just give them a thorough weeding and they should be good to go. Can't wait to see the rest of your pics.

Could be Croton capitatus or one of the other common Croton's known to grow in Texas. Here is a link that might be useful: Wooly Croton

It looks like willowherb (Epilobium ciliatum) that has finished flowering and is setting seed to me. Did it have very tiny pink flowers? If so - Weed. You may have accidentally planted it, can commonly come as seed or seedling in nursery containers when you purchase plants. Grab it before the seeds are dispersed, pulls easily.

Sod is often attached to (actually grown through) a plastic netting as it hold the rolls or sheet together during transport and installation. Hydroseeding is just spraying in a slurry of grass seed, fertilizer and some sort of cellulose stabilizer - no netting involved. You want raised beds to be in contact with the underlying soil - no gravel or weed cloth. Smothering the existing growth with overlapping layers of cardboard is easy and effective and it will decompose in time, allowing the roots of the raised bed plants to penetrate down as deeply as they like. Using a prepared herbicide according to label directions is really the best way to approach weeds in the lawn, particularly if there are more weeds than lawn. In fact, if it is really bad, razing the entire lawn and starting over may be the simplest way to go. As long as you don't spray any herbicides (plant killers) directly on what you are going to eat (and if you think about, it that makes no sense anyway!!), then there is really very little concern. Most of these products breakdown in the soil quite rapidly with rain/irrigation and the activity of the soil biology into harmless elements and they are not taken up into the plants or fruit. Weeds in dedicated planting beds that already contain valuable or desired perennials and shrubs are best handled with manual removal to avoid damaging the desired plants - herbicides can't tell the difference between a weed or anything else that might be growing :-) If you do not intend to add any plantings in these areas right away, then smothering with cardboard will work well too, at least for most weeds. Cover the cardboard with a good thick layer of mulch to keep it damp and hasten decomposition. And mulch any other bare ground after removing any weeds. Mulch is about the best weed suppressor there is and is of multiple benefits to the garden.