You can cut off most of that and it will put out new growth. Just know that you will likely disrupt fig production depending on how much you cut i.e. chainsaw. Ideally heavy pruning should be done in winter/dormancy but you aren't going to kill it outside that time.

http://forums.gardenweb.com/discussions/1893495/fig-tree-pruning

ID usually requires the fruit. Some areas can only grow certain varieties problem free so it should be easy to find out which it is. Try asking around other locals with fig trees.

I agree with A J also, All the suckers at the bottom should be pruned away

and I would recommend that you cut off a few of the larger branches that are facing right up into the sky, instead of snip snipping here and there

This will reduce the tree height and you will still get some figs

you could spread pruning over the next couple years and will eventually renew all the wood in the tree to the shape you want

Then again figs are very resilient so you could literally take a saw to the whole thing about 2' off the ground and it'll grow back nicely

It survived -10 a few winters back. I'm 5'5" and the suckers to the far right are a few inches over my head. Also, the vast majority of the suckers, I kinda tangled them up on purpose for the time being. I kept either getting slapped or poked while trying my utmost to rip the ivy, blackberries roots and all and clear out other accumulated debris from the area. I will untangle when I finish clearing the ground around it.