Mulberry Tree in Las Vegas, NV

Hey everyone, looking for a little advice. I just bought a house that has a Mulberry Tree in the front yard. It has a few dead branches but appears to be pretty healthy overall. There is a lot of fruit currently on the tree as well. Here are some pictures: http://imgur.com/a/nI4NX.

So I'm looking for some general care and maintenance advice. Is it ok to cut off the dry branches? How much should I take off?There is currently no irrigation system set up for the front yard, so for the past 2 days, I've just left the garden hose dripping at the base for a couple of hours. What type of irrigation is best for this kind of tree.

Thanks for any advice you can share!!!