Trying to replace a wheel
First off, the number one cause of broken spokes is not enough tension. When spokes aren't tight enough they load and unload with each revolution of the wheel – basically they are getting bent back and forth each time they go around. Over time they break, just like bending a paperclip. The most likely place for the spokes to break are at the bend at the hub.

Any good wheel builder knows this and will take great care to get the spokes uniformly tensioned. Once the spokes are tight enough they don't load and unload on each revolution and they will last a good long time.

Here's what you need to know to order a new wheel:


The size of the wheel: 700 C on your bike.
The rear dropout spacing: you can measure this by taking out the rear wheel and measuring the distance between the dropouts (the part of the frame where the wheel attaches) in millimeters. Odds are good it will be 135 mm, although 130 mm is also likely. They need to know this to choose the right hub width.


Beyond that the wheel builder should be telling you (or at least making recommendations). I think the bike shop questions were being asked to learn about how you ride and care for your bike.

Unless you've been riding hard (jumping curbs, bouncing through pot holes, etc.) I'm very suspicious that your spokes were too loose. Really, you're not "too big" for a decent bike. Trek, for example, says their road bikes are good for riders up to 300 pounds, and mountain bikes are good for 275 (if I recall correctly). I'd guess that the mountain bikes are lower due to the rough service they are likely to get.

Some things I'd expect your wheel builder to recommend:


Don't use heavier spokes, use more. I'd go with more (at least 36 or even 40). On our tandem our wheels are built with 40 14-gauge double butted spokes. They are holding up just fine, and our team weight is 360 pounds, with the bike and bags the load is up around 425 pounds. We're using nice sturdy rims, but the big thing is that I built them with good tension on the spokes.
Use the biggest tires that will fit and run them with relatively low pressure. This will enable the tire to absorb road shock, and being big it let you run a lower pressure without risk of "snakebite" punctures in your tubes.
Consider using an asymmetrical rim in the rear. Asymmetric rims offset the spokes to the left a bit to help equalize the bracing angles of the spokes (on a symmetric rim the drive side (right) spokes are much tighter because they are at a steeper angle. This is because the flange of the hub is moved to the right to make room for the cassette.