Tangled vines and dead wood

Hi,
I've recently moved into a new home with a small garden. Much of the space is bricked or gravel but there are several well established vines that have become a tangled mess. 

Can anyone help identify what plants I'm dealing with here, and then ideas on how best to prune it all back for a good show next year?

I can see three different climber/vines that have been deliberately planted on the inside of the garden wall (brick with now broken trellis on top).

When I moved in a month ago there were some large and attractive red leaves that have all fallen now. What is left is a green tri-leaf that I'm seems to be strangling everything else.

Before moving in, a 'maintenance man', seems to have hacked a lot of this vine up along the outside wall which runs along a pathway. He might have made a mess of it.

Generally the tangled vines appear to have a lot of dead-wood and the weight of it has broken up the wooden trellis in places. As I'm renting I'm not really worried about replacing this. I'm happy to put some work into tidying up though. 

Any thoughts and feedback greatly appreciated. 

Joel