You are a little heavier than I was when I started, but also a little taller. Height won't be an issue, any bike shop will have a bike in your size. As a starting point I'd suggest trying about a 56 cm frame, but they will be able to guide you better based on your proportions.
Weight should only be an issue on wheel durability. In your price range there are a lot of options, but all are going to come with inexpensive wheels. If they are properly trued and tensioned, they will be OK as long as you don't start jumping curbs and avoid potholes as much as possible. You're going to need to true them more often than a light weight rider, but as you ride, the weight will come off. I put maybe 2,000 miles on a set of DA-16 wheels that came on my Specialize Allez before I started breaking spokes. By 3,000 miles I was really eating into the braking surface anyway. Wheels will be the first upgrade you need, but not right away.
As for selection, Trek, Giant, Specialized, Felt...there are many brands that offer entry level bikes starting at about $700. Test ride what you can and choose based on your comfort and what dealer is available to support the purchase after the sale. Visit bike shops in your area and ask questions. Don't let them sell you a hybrid or comfort bike.
Using cycling to shed pounds is a matter of regular riding and modifying your diet. Nothing much different than any other method, diet and exercise. Long slow distance is the key. A 12-13 mile ride isn't going to do a lot to help you burn off fat, you generally need about 20-30 minutes to get past burning the calories readily available and start burning fat, so if you can use your possible commute as physical conditioning and then go into a longer ride on the way home or afterward, you'll start seeing results. A general rule for you at this point will be about 90 miles to lose a pound of fat, but if you get into the active routine your results will be improved. You will hit plateau where you seem to quit losing weight, you just have to work through it. Watch the fat and processed food intake and it'll all come together.
The training and nutrition forum will be helpful, and the Clydesdale forum will provide lots of encouragement. Good luck, and welcome!
"It hurts so good..."