This will be my first time traveling with my dog. He's not big, only about 14 pounds, but he is longer and taller than most, about 18' long and 12" tall. After doing a lot of research, I'm trying to avoid him going into the cargo hold at all costs. I bought the SturdiBag XL which he'll fit in nicely, now it's just getting him (and therefore, myself) on the plane.
It's only a 1 way flight direct from Houston to San Francisco (relocating) and my company is paying for the costs, so that's not an issue. Is my best bet to fly United First Class (it's either the 737 or the A320) and just hope/convince people to let him on if there are any issues? Is there another airline that may make it more doable? I was thinking First Class would provide the most room, although the foot room is often divided by 2 bars (so into thirds), is the window seat my best bet?

Separately, if I reserve his space with United online, do I have to get him checked anywhere in the airport (besides going through security)?

I appreciate any and all advice, it's been hectic trying to find the best solution.