Hello. Yes you can prune in the summer but it's not ideal, the best is when the leaves start to fall. If you do it in summer do it progressively one branch from time to time to not stress the plant too much, also wait for a cool week to not promote fungal infection.

You can prune it 50%. Don't just cut a leveled line, cut it leveled on average but cut different stem at different length or all the new crown will happen at the same height and you will eventually end up with a table surface of wood over time. As the plant seem very tall, some trunk should be lot larger than other, remove some of those. When something got out of hand over some years, don't try to remedy to the situation in just one year, spread the big trunk pruning on 3 years. Just to give you an idea of how much you can prune, watch this:
http://www.youtube.com/watch?v=BIT8K6Yxqf0
Note that this is a healthy one, don't go as far as this if you have lot of dead wood on the back, that is why i mention to first remove 50%, allowing for light to go down the plant and once new growth are confirmed to restart down there, you may remove some more.

The back may indeed be dead because of the lack of light that you mentioned. It may also be the lack of water for the roots section under the roof or lack of aeration causing fungal disease. But my bet is on the lack of light.