The bike that fits that description best is the Birdy. Great loaded touring bike, good multi-mode, fast, comfy, durable (mine is way past the 10,000 mile mark, a fair amount of it off road loaded with panniers). It's my only bike and I use it for similar purposes. One downside is that it can't be rolled when folded (unless you buy a rack in Japan or Europe with wheels). Thus, I get yelled at when I take it into the grocery store unless I carry it or fold it put it in the cart. The 8 speed is in your budget range, but for 24 speeds, you'd need a used bike.

It easily fits my 5'2" girlfriend with a lot room to go lower, but you carry around a lot of unnecessary seat post.