At under $200 each, I'd go straight to craigslist for used bike. You will get a much better quality bike than any department store bike and better than you can buy from an LBS at that price point.

Probably the best for your use is a used, vintage mountain bike (one with a rigid fork) and/or a used hybrid. Both have comfy fat tires and they tend to be less expensive than an equivalent quality road bike. Vintage road bikes generally command a premium.

This pretty much answers most of your questions since you will be riding whatever comes stock on the used bike. You may want to swap out the saddle on the bike you buy. You will want to swap out the knobby tires on an MTB for slick tires. You will need to learn how to do some minor adjustments on the bikes. You will want to buy a pump with a gauge since proper tire pressure is very important.

Insofar as size is concerned, there are a number of rules of thumb that you can read about online. As long as you can straddle the bike comfortably with 1-3 inches of clearance you should be fine.