Sharbear, Your elevation and the slope of your yard may also have something to do with it. If you're in a low spot for your neighborhood then you'll need less water than your neighbors because all of their excess is draining towards you. And even just a few centimeters can make a difference in spots. I have two plumbago bushes just ten feet from each other in identical light and soil conditions but the slight slope of the yard means that one gets too much run-off and will never ever be happy. (Guess I should rip it out, but the list of shoulds is long in my yard.) Another thing you should consider is a septic tank. Do you have one? Did the house have one at some point? If you only have dollar weeds in a certain area, it might be a septic tank that is causing that area of the yard to drain differently from the rest of the yard.

Most likely it is "Poa annua", annual bluegrass, This is a cold weather "weedy" grass that will die out in a few weeks. It is often used as a nurse grass in seed mixes since it does germinate much more quickly than most others. Most often, in established lawns, the seed will germinate and start to grow in late August and most all pre emergent need to be in place about 6 weeks before, so usually mid July is when to put them down.

I don't know that plant at all. but if the ID is correct, it's goji berry, a currently popular fad food/juice for anti-aging.

It looks like you have a bermuda lawn. Do not use Roundup, it will kill your lawn. You can spray the entire with a broad leaf week killer. You can get one that attaches to your hose. You may need to spray it a couple of times. That will take care of the weeds that are not grassy weeds. For the grassy weeds it is best to prevent them with pre emergents. They are granular and are applied like fertilizer. If you are in N. TX then you will need to apply that on March 1, June 1 and Sept. 1. If you live in other parts of TX then check with your County extension agency for the proper dates. Keep your grass cut short, 1-1,1/2" . Taller grass will not be as thick. Mow low and mow often. This will help to thicken the grass. Fertilize once the grass start growing well, lightly every 4-6 weeks if you are trying to improve your lawn. Good luck!