You can tell if an oak is a Live Oak b/c it is evergreen. Not sure, but I think it is the only oak that keeps its leaves year round. Those trees are awfully close to each other and you're right about their roots sharing the same space. Oak roots spread out. I am always surprised how far the roots of my Red Oaks go. Anyway, I think you should google Oak Wilt Disease. It affects Live Oaks and Red Oaks (makes me paranoid). It can spread for blocks. In the late 80s or early 90s Austin, TX had an attack of Oak Wilt Disease and lost untold numbers of trees.
Is your arborist a certified arborist? And what is he proposing to do for $700? What, specifically, did your local fertilizer dealer recommend?
You definitely have some dead branches. And where there are the rust colored area, can you see any insects?
