Fort Worth is under construction right now. I35, the main north/south corridor has been two lanes each was for at least the 30 years that I have been driving. They are putting in more lanes, however they will be toll lanes. If it's anything like 820/183, the east/west corridor, it won't do anything to alleviate traffic unless you are willing to pay the approx $7 each way. What this means is don't look at distance for your commute, look at time. I live in N. Fort Worth and used to work in Irving, 26 miles. It was normal for it to take well over an hour in the afternoon, especially Friday afternoon. However, 26 miles the other direction would be about 30 minutes. I don't think you will find a 15 min commute to both downtown and Westlake.
$350k will get you a lot of house around here. Mine is 2300 sq ft in a nice neighborhood, Keller schools, built in 1995. Currently valued at $225k, taxes are $5200/yr. I would really recommend renting for a year and learn the area before you buy. Just a few miles in any direction can vastly change your commute time. It would suck to buy a great house in the wrong area. Haslet looks good on paper, looks good on a map. Affordable houses, any store you can think of, good schools, everything is close. However, due to all of the two lane roads, it is an absolute whipping to get around during rush hour.
714 Main is an XTO building. They are moving a lot of the company to Houston in the next year or so, something to keep in mind if that is who you will be working for.
This is a car city. Anything area older than about 20 year is going to be sketchy. Narrow roads with little to no shoulder. Google maps does a pretty good job of bicycle routes, I would plug in your destinations and see what it comes up with. If there are not dedicated bike lanes/paths, it will try to route you through neighborhoods. I cycle as much as I can, but am limited to about a 5 mile area west and south, 12 miles north and east without having to travel on very busy roads. The good thing is just about anything I need is within 5 miles. Except for a good bike shop.
I've lived in the area since 82 and would be happy to answer question or show you around.