Can you identify these 2 lawn weeds and how to control them?
A tough question and a good one. Your last two pictures are Sedum Acre, a particuarly vexing weed to control, but easy to identify. I believe that the second weed, your first picture is Scarlet Pimpernel (Anagallis arvensis) another hard to control weed because of it's growth characteristics. That being low growing and seed producing. It is often confused with chickweed. Scarlet Pimpernel is a member of the primrose family.
To try and control these guys is another situation. First off, your lawn grasses should be kept at the 3 to 4 inch height for the purpse of denying the new seeds the sunlight to start growing. With that in mind (seed growning into adult plants) you need to lay down a pre-emergent product like Preen to stop the seeds from developing further. The next step for both of your weeds is a product call Weed-B-Gon Purple Label. I strees purple label because the chemical really does work on broadleaf weeds very well.
With these two you are going to have to be persistent. Reapply in the fall, late September or October, when the plant are trying to set down new roots for over-wintering and next spring when they are first flowering. I see that your sedum is flowering now so that is good for controlling it. With the sedum pulling is easy but darned hard to keep up with because the fleshy leaves keep falling off and producing new plants by themselves.
I hoe that I have helped you with your question.
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