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It is best to prune an olive tree anytime between the end of winter and before it starts flowering. It is best to make sure to leave the tree alone until it is about 4 years old. After this you may prune the tree. Here is an article for more information: https://www.gardeningknowhow.com/edible/fruits/olive/pruning-olive-trees.htm

It seems that olives were actually grown commercially in South Carolina in the 1800s, so your climate should be fine for them. It could indeed be a problem with the varieties needed to cross-pollinate. Is the tree producing flowers? If not, first, see this article to check whether there might be any other problem:
https://www.gardeningknowhow.com/plant-problems/environmental/plant-not-blooming.htm

These articles have information about the varieties needed for cross-pollination. You may also want to ask a university extension agent whether there is a way to check what variety you have.
http://edis.ifas.ufl.edu/ep515
http://ucanr.edu/blogs/blogcore/postdetail.cfm?postnum=21374

There should be no reason not to plant them in the same area.

I did a bit of research and learned that in London gardens they often use Olive Tree's to build perimeters around the vegetable gardens; they are planted much like you would see Boxwoods.

I do think that the lack of water had a large part to do with this. They will often require irrigation for excessive dry spells.

It could be that it is not a self fertile variety. In this case, you will need another variety to pollinate with. Most true olives will not require fertilization so this will not, usually, be the issue.

This article will give you more information on growing olives: https://www.gardeningknowhow.com/edible/fruits/olive/olive-tree-care-information.htm

The only thing to be done is to prune out any dead growth and wait for recovery. Just leave any living growth alone and it will recover with time. This article will give you more information on the care of olives: https://www.gardeningknowhow.com/edible/fruits/olive/olive-tree-care-information.htm

This can be due to several different types of insect, and some fungi.

I would recommend applying neem oil to the tree on a cloudy day, to all of the portions that you can reach. Then apply wettable sulfur, and dolomitic lime. These will get rid of anything in the soil, while conditioning it and providing a few nutrients for common deficiencies.