I believe the tree nearest the house is a bradford pear, which apparently has a host of issues on its own. It's definitely pushing the walkway up too. So that one most likely needs to go but now that I look at it, it doesn't seem to be the one causing the shading issue.
The one on the street strip is overgrown as well, hopefully some aggressive pruning would let me keep the tree and mitigate most of the shading problem.
BrettS did a great job of laying out the pros and cons - the NE roof may be just as productive given the shade issues. I see a garage in the picture - can that roof be used? Maybe use the SW roof on the garage and the NE roof on the house. Getting power to the garage may be more expensive, though.
I was thinking about the garage as well, I think I could fit just under 6kw of panels on there. Unfortunately it's on the far opposite corner of the house from the main panel so I'm assuming that would add a fairly significant cost to the install? ie a long conduit run, part of which would need to go under a patio/walkway. There's no subpanel in the garage, if that makes a difference
@AnotherMike, Looks to be a California Pepper Tree (Schinus molle).
I ran the tree near the street through the PictureThis app (awesome app for identifying plants btw) and it tagged it as a Chinese pistache tree, which looks right based on a quick google search