Invasive ground cover
Thank you for your question. Your photos have three plants that many gardeners (and horticulturalists) call "weeds." The one on the sidewalk is moss. The one with small round, green leaves is speedwell. The variegated-leaf one is either Lamiastrum ((yellow archangel) or Aegopodium podagraria (Bishop's weed or goutweed). Both of the latter are sold as groundcovers and can easily become invasive. I'd need to see the leaves and stems more closely to distinguish, since they both come in varying degrees of white/green coloration. What all of these plants have in common is that they prefer shady, moist and cool environments.
The speedwells: http://hortsense.cahnrs.wsu.edu/Search/MainMenuWithFactSheet.aspx?CategoryId=6&ProblemId=2038
Yellow Archangel: https://s3.wp.wsu.edu/uploads/sites/2054/2014/04/Yellow-Archangel-brochure-2015.pdf
Moss control: http://bryophytes.science.oregonstate.edu/page18.htm
In the long run, so long as the environment in this area remains the same, so will the weeds and you can look forward to many years of ripping out, spraying (with only limited success), and scraping. Moss is probably always in your future. Avoiding planting the other two and vigilance may reduce your work. Good luck!
Thanks for your helpful response! I thought my photos did not make it through, so I didn't look for your answer. Just found it today - March 1st!
I'm glad you are there and volunteering your expertese!
Another expert says it’s liverwort, not speedwell.
Wow, there are a LOT of varieties of liverwort. Posting new pix. The patch is 3" x 6". It is very compact and low-growing. I hope the closeup helps.
Thank you for the clearer photos! The other expert has not weighed in on which variety, but I have the ID out for examination. But, control of one is control of all: https://oregonstate.edu/dept/nursery-weeds/weedspeciespage/liverwort/liverwort_control.html