The further down you probe and cut out, the longer it will take for them to re-appear. Perhaps you can estimate a straight line from original to runner sprout and chop off the root of the runner on the inside of your original raised bed. That would remove some energy but would not prevent the remainder of the runner from re-sprouting unless you can fish it out of the ground from beneath two raised bed sides. Usually repeatedly removing sprouts at ground level makes this activity go away by mid summer, and of course it returns next spring.

First 5 shots show a Rose... Which variety is anyone's guess until it blooms. It might be the invasive mulitflora rose -- of which there are two varieties, one blooms pink the other white. Watch for flowers and post them if you're still having trouble ID'ing the rose -- there are a few non-invasive varieties with this wild looking growth habit, so I don't want to label it as something to be destroyed unless I'm sure. Granted, some people want to keep them even if the invasive variety 'cause they are pretty when in bloom. Shot #7 shows a berry --- either black or raspberry. New growth generally looks fairly similar between 'em, but mature leaves and blossoms can definitely rule one out. Whichever, if you don't mind them, they can be welcomed food for wild birds and critters and are edible to people, too =) The last shot isn't a good angle to ID anything -- sorry...

You might get some clues about the gall (or whatever) by cutting it open. It looks like whomever was there left already...I'm just gonna go stand over there when you do it :) If it's a blackberry or similar the correct plant forum here might know what it is.

BW is a thing you'll curse and hate forever, don't you dare take a bag of earth from there to use anywhere else in the garden. That's pretty much the only thing you can't kill in this lifetime in that area. And that black earth you just added prerhaps for a vegetable garden, is unprotected from it. Maybe you can still dig in a wall of at least 20 cm deep between the weed and your planting bed. For the vine, don't worry. You can always cut it, just wait until it's in leaf to decide if you keep it or not. Give your garden a summer without intervening - it will tell you what grows where, where is the right place for planting stuff. That hedge might protect you from scorching sun or from your neighbour's sight. Next winter is time enough to plot and plan and modify to your heart's desire.