1) What is the definition of a weed? Does it fit that definition? If a weed looks good, do you necessarily want to get rid of it? Some are really aggressive and will grow like crazy and reproduce themselves lots too. Maybe some are ok though.

2) From a horticulturist perspective, a weed is a volunteer plant growing where you don't want it to grow. If it's attractive and isn't aggressive, like LimeyGeorge says, go for it! If it's a noxious weed it will either reseed like crazy (you've already had it growing for a season? and haven't had others?) or it will send runners out and sprout up everywhere.
The vine looks nicer than a morning glory, from your picture, with interesting veining in the leaves? Why kill it if it looks nice and isn't taking over your yard?
The second picture looks like the plant could be hesperis matronalis/Dame's Rocket. (the seed pods are NOT connected to the bulb-like foliage on the far right, correct?) Did it bloom with fragrant blue/purple flowers? It can naturalize by reseeding, and is considered a wildflower. It can fill in a spot nicely in a few years, but is not considered invasive - not in NC, anyway. I think stheinz had a picture of it in his yard somewhere. Check it out and see if it looks familiar.

It's pretty difficult to tell for sure from what is left of the plant.

3) When I have time I will post a picture of a weed that I just trimmed up and let grow because I like the way it looks. I don't kill wild violets and I think dandelions are pretty.As far as my lawn goes as long as its green I just mow it-I like clover and so do the dogs. I truly think its up to you-if it is pretty and not invasive-keep it.