Why? I doubt that he's leaving the Texans, but even if he does, what has he done? Has he developed one QB? No. His Texans teams have woefully underachieved.
One or the other. I don't think one guy can wear two hats successfully. If I remember right, Shanahan was HC/GM for the Broncos and they didn't do well at that time. He certainly has the knowledge and experience I would want in a Pres. of Football Ops, but does he have the temperment - is he too hands-on and get mired in the details and lose the ability to keep focused on the big picture? Or, as HC, is he too old and burned out, and would he be at cross purposes with whoever the GM or Pres. of FB Ops is? I think there might be better fits, but I wouldn't rule him out completely.
Bill O'Brien would be a fine choice as I do believe he is a very good HC, I also think he may just be the most realistic possibility out available come the offseason. I'm excited to see how Watson develops under his tutelage as he is the only young QB with any talent that O'Brien has had a chance to develop. At this point he wouldn't be my first choice, that is reserved for Payton but I do think he is more realistic of a possibility than Payton is and would certainly be on my short list.
Belichick, Pete Carroll are two who have done it successfully in recent memory. Andy Reid currently has that role. Fisher failed but he always sucked as a coach. In Washington Shanahan battle with Bruce Allen and Daniel Snyder, most recently over the RG3 stuff. Turns out he was right about Cousins being the guy all along.