Cyclocross

First issue is frame, fork and brake clearance. CX tires are considerably wider than road tires, so a nice and slick road bike usually won't have room for them. But if you have friends who run CX it's easy enough to find out - borrow a set of wheels, try to pop them in and see if they fit.

The issue about CX wheels is a little silly. Wheels used for CX are usually built to be more rugged than wheels solely intended for the road, but then I don't know what your road wheels are, and how heavy/strong you are. Trying to run a 18/20, or 20/24 combo as CX wheels I would consider as flirting with disaster, while a 32/32 touring set would probably stand up well even to CX (assuming a "normal" rider weight).

You'll probably find that your gearing isn't ideal. But what you have should work good enough for you to find out if you're willing to spend more money on it or not.

Interrupter brake levers can be installed at a moderate cost, if you want them.

+1 on dabac's reply.

The key thing is going to be tire clearance. Cross bikes are typically outfitted with cantilever brakes, sometimes V brakes and disc brakes are gaining popularity. In all cases there's a ton of clearance in the stays and below the fork crown to allow for big knobby tires. Standard road bike caliper brakes don't have anywhere near the clearance although some long reach style brakes might work on narrower cross knobbies. I've managed to squeeze 700x28c knobbies on my stock road frame for an unusual mostly dirt event a couple of years ago but knobbies that narrow are tough to find.

I run stock road wheels on my cross bike, I train on normal road clinchers with 16 bladed spokes up front and 20 in the rear with no problems. I race on Easton EA70x tubulars with their 24x28 spoke counts again no problems over a couple of seasons of racing but I probably wouldn't run low spoke count carbon or ultralight rims.

Gearing is typically lower as dabac mentioned but not necessarily all that low. I run a 46 big ring and 39 small ring coupled with an 11x28 cassette or even an 11x25 cassette for racing if the course doesn't have sustained steeper climbs that are rideable. It's not mountain biking so if the climbs are steep we're typically running. Most folks run compact cranks with something like 46x34 rings.

I really think your challenge is going to be tire clearance. FWIW, you can pick up used and very good cross frames or complete bikes for a song on Craigslist or ebay. Even if you don't like riding in the grass and mud they make awesome winter riding and commute frames with tons of fender clearance and slightly relaxed geometries.

Good luck,
-Dave