Shimano Bottom Bracket LHS cup - splines all worn off. Removal tips?
Q1 - Removing the damaged left cup: I think I'd do this by first removing the crank arms and the right hand cup. Then, I'd remove the axle. Once you have it all apart you can work on removing the left hand cup. Since it has a right hand thread what I think I'd do is to get an screw extractor (aka an EasyOut). You can find sets online fairly inexpensively, Harbor Freight also carries them, and your local hardware or plumbing supply store will also have them. If you're going someplace local, take the right hand cup to make sure you get the right size. With the EasyOut you'll be able to back the cup out using the hole for the axle.

Q2 - Can you buy just the left cup: I'd be surprised if you could, but bottom brackets are relatively inexpensive. A descent conventional bottom bracket lists for US $30 and you can find them pretty easily discounted to about half of that. It seems likely that enough parts of that bottom bracket are damaged that it would be worth replacing the whole thing.

Q3 - Should you use a thread locking compound: You could, but it's probably unnecessary. The left hand thread is self-tightening as long as the bearings are doing their job. It's a bit counterintuitive at first, but think about how the bottom bracket works. From the right hand side the crank is turning clockwise. This causes the ball bearings to turn counter clockwise, so the motion of the bearings on the outside – against the bottom bracket cup – is working on tightening the left hand thread.