USA1 wrote:I would avoid 3, 4, and 5. You don't want to live near anything that ends in "ward"
On a more serious note, the housing bubble is alive and well in Houston. Bare lot values near your work are over the budget you posted, so I can see why you're looking where you are. Binz (5) is close to Montrose and the Museum District, and within 2-3 miles of UH, TSU and Rice. There are iffy areas but that's true anywhere. No "safe" neighborhoods, right? Technically part of downtown (4) that side of 59 was one of those places where we wouldn't go at night, but they're building a lot of guarter-mil homes and condos there now. Who knew? The Heights could have been a good investment 10 years ago but prices there have boomed and spread to its fringes (1, 2, 3) where you'll see expensive new construction near shotgun shacks. Sometimes in the same block. The Heights also has those wonderful tracks running through if you like the sound of train horns.
One thing I suggest is going back without your real estate agent and drive around the neighborhoods on your own. Maybe do a spiral search pattern out 5-6 blocks from any house you're considering, to get a feel for what's nearby. Only you can decide what feels safe to you.