1. I call this celery weed. 2. They look decent to me so I might keep them. 3. There might be a few similar ones like this. Let me know if more pictures needed. 4. This looks good so I'm hoping it's a flower. They came out with grape hyacinth in a flower bed.
#1 and #4 look like something in Ranunculaceae family, could be Ranunculus or Delphinium #3 looks like a weed, don't let it go to seed.
Yes #3 is a weed. But it's not the ugliest ones and it has shallow roots. #1 I think is weed too but hope #4 some flower.
Yes, it could be Ranunculus repens, there even is something what looks like a stolon in the top-right corner.
#3 is probably Hairy Bittercress, Cardamine hirsuta, which is an introduced weed; however, it could be C. oligosperma, a native plant.
4. Looks like Eranthis hyemalis. eranthis hyemalis - Google Search Flowers will be over but there may be some seed heads to be found. See... Photo of the seed pods or heads of Winter Aconite (Eranthis hyemalis) posted by bonitin (All Things Plants)