The top one is spotted spurge, a common weed. Scott's website says that weed b gon will work, but so will pulling it up - make sure you get the roots.

Both look like spurges (Euphorbias.) E. maculata on top, the bottom may be E. supina. Both have latex sap, might want to wear gloves for pulling. They are annuals that drop a lot of seeds. Pulling before they do that will dramatically reduce the number next year.

Both are native in continental US.

Easy to remove by hand. Has central tap root.

I've been pulling them (without gloves, I will start using gloves from now on), but they are spreading faster than I can pull them. I've sprayed the one on the mulch areas with weed killer, but can't do that to the ones that are now spreading on the grass.

I'm going to try Weed B Gon Max Plus Crabgrass Control as recommended on Scott's website.

There may also be discussions about spurges in the lawn in the lawn forum. Their milieu is hot, exposed, dry spots. If an option, watering (more, or at all if none currently,) letting grass get a bit taller could help eliminate it from lawn areas. Keeping a layer of moist mulch/pine needles/leaves, organic matter of any kind, will help reduce its' affinity for bed/non-lawn areas.