Cyclocross bicycles make excellent multi-purpose rigs. They聮re sturdy, sometimes come with hill-friendly compact chainrings, have room for fat tires and fenders, and may even have braze-ons for mounting racks and the like.
The ConquestI use a Cross Pro ($2,000) as a commuting/rain bike. You can get them at Performance bikes for not much more than your top end price, and the slightly lesser Cross Comp (lower component level) goes for about $1,300 retail. Aluminum frame, but it聮s a nice-riding bike (and I share your predilection for steel) and has all the stuff I mentioned above. So it could tour as well as commute, trail ride, etc.
聮s Conquest (about $1,000) also has an aluminum frame. But it聮s a very popular bike that can be adapted for just about any purpose. It even has disc tabs if you want to switch out the canti brakes for discs. Shimano and FSA components are sort of low-middle range, but they聮re perfectly suitable. And the frame warrants component upgrades in the years to come. Add some fenders, and the stock bike with its 36-46 chainring (for a little more oomph on hills than straight road chainrings) and fat 700×35 tires will take you anywhere you want to go.
聮s venerable 520 is a steel touring bike, a real load-hauler. Nine-speed cassette, triple-chainring, braze-ons everywhere. It聮s a bit of a pig, but a very rugged bike and perfectly fine for all the things you want to do on a bike. Retail is $1,300, but you might find one on sale. Or, on eBay.
Keep the rubber side down!