First question: There are poison that can be applied specifically to the plant you want killed. There is no reason to spray the area. However since it has been growing for years in the same place you may have to take drastic measures to get rid of it.

Second question: You can not take a tree out, unless it is only a couple of years old. What you can do is grind the stump down below ground level and fill in the hole.

When the stump was ground out did they remove all of the wood chips and fill with good dirt? If not over time these wood chips will rot.

Even if you removed all of the wood chips from the ground down stump, there are probably substantial roots that the grinder could not reach, They are now rotted to the point the ground has subsided.

Yes I hate that fungus and sunken area. I have such an area in my yard also. I even asked my lawn care company if they can help get rid of it (they do care for my lawn, right?) but they said, if it's there, it's because some conditions are good for it to be there, so they can't help with removing the fungus. I said, yes there used to be a tree there. The answer: Well there you go, that's why you have a fungus growing there.

As for the extreme option of tilling up an area with weeds -- yes go ahead. Get rid of that initial area first. New weed seedlings that pop up -- get rid of them also. Keep it under control and you can reclaim it.

As for the ones near your valued plants, you may have to do the manual option and just go in and pull them up. At first it will seem daunting but keep at it and you'll at least get the upper hand on the ones near your plants.