Alliaria petiolata, Garlic Mustard. Here is a link that might be useful: Google image results

I don't recall the name but yes, it's a perennial vine that can be quite persistent. Dig and pull it every time you see it in order to eventually starve out the root. Even if you use weed killer, it will still take considerable time to do it in, this because of the sturdy persistent root. In general, Roundup is applied when a weed is vigorously growing and either in bud or just beginning to flower.

Found! It's a Vicia acutifolia, commonly known as Four-leaf vetch (even though it usually only has two!)

I'm not sure that that's chickweed. I'm not sure what it is, but it kinda looks like a native wildflower, Venus' Looking Glass (Triodanis perfoliata). Have a look at these photos at the Lady Bird Johnson Wildlower Center to compare. Chickweed is surprisingly tasty, but I'd check out your plant a bit more before eating it... Here's a link on foraging for Chickweed, and the person who wrote the article mentions that it is sometimes confused with Petty Spurge. Weeds grown in pots in favorable conditions may look a bit different from weeds grown out in the yard, competing with grass and other weeds... Good luck, Virginia