Fuel oil isn't necessary. Paint the newly-cut stumps with glyphosate (Roundup) or triclopyr (Brush B Gon) -- it's better if it's the concentrate rather than the spray. It doesn't matter if it rains later: it's those first few minutes when the still-living tissue is in contact with and soaking up the glyphosate or triclopyr that matters. Once the glyphosate/triclopyr has dried or the top layer of cells in the stump has died, no protection is needed.

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.

In the first picture it is called a dewberry, close relative of a blackberry. Flavor varies, usually as good or better tasting than named varieties but not as productive. They fruit on second year canes. The first picture is of a second year cane that will produce small berries. Since it is in shade probably will not be of good quality. The second picture is first year cane. If left to grow if will produce flowers next year. They are edible, but will continue to spread usually by underground roots. Best off to pull, with gloves. If kept pulled will eventually starve the roots and will not be as prolific.