Possible? Absolutely. Practical? No.
Above is a picture of a ship of the line which was more or less a common sight in warships in the 1700s. Those cannons were heavy, ranging from 450 to 1000 pounds. And most ships of the line carried dozens of them. That doesn't include the powder and cannonballs to operate them. So depending on the size and weight of your dragons, that shouldn't pose a problem for the technology level.
The runway could pose an interesting issue, mostly because of the lack of sails. But, I'll get to locomotion later. The real issue here, is the material you'll use. If these dragons have claws, wood is a poor material to use for the runway. They'd scratch and claw at it, rendering it useless before long. That means the thing that they need to run on will eventually be little more than wet sawdust. I'd recommend a layer of sand or dirt so their claws can sink into it without requiring expensive repairs.
As for the now lack of sails? Oars were still common in the age of sail, for smaller ships and lowering the crew required to travel. Because oars were not quick and certainly were not efficient.
However, galleys were known for being oar-propelled. There's an issue, though.
The galley is characterized by its long, slender hull, shallow draft, and low freeboard (clearance between sea and railing). Virtually all types of galleys had sails that could be used in favorable winds, but human effort was always the primary method of propulsion.
The two concepts--ship of the line with its multiple decks, and the galley with its oar-power--are mutually exclusive, for the most part (some exceptions have been brought to my attention). However, you literally carry your own answer. If you were to combine oars with dragons pulling like a draught animal? You could still achieve a decent speed. How good? Depends on how many oarmen and how many dragons (and how much power either can offer when contrasted with the calories they'll burn).
There is a unique advantage to having dragons as well. They can fly in more food to prolong time at sea. 308
Granted, that means it'll be easier to track them, since they'd need regular food runs, depending on how many pounds of meat the dragons will need per day. But it offers the possibility.
However, food runs are limited by distance from the shore (or food stocks) to the ship. And there's also the limiting factor of carrying capacity for the dragons. This could be mitigated by fishing, but these are warships--they aren't designed to focus on such things, normally.
That being said. Is an aircraft carrier the most practical application? No. They'll be slow and cumbersome ships with incredibly high maintenance cost. The best I can imagine this would be used for is hit and run tactics near the coast, and that's kind of defeating the point of having dragons that can fly and land anywhere there's a stretch of land.
Why wouldn't an aircraft carrier for dragons be useful? Average speed of a ship of the line? 12 knots. That's 22 km/h. And that's with the slew of sails meant for propulsion. Average speed of a galley? 3-4 knots (5.5 - 7.5 km/h), but 2 knots (3.7 km/h) in formation. What's the average cruising speed for your dragons?
Say a dragon's speed is comparable to a bird of prey?
* Bald eagle: 120-160 km/h
* Golden eagle: 320 km/h
* Red-tailed hawk: 190 km/h
"Well, yeah, but isn't that the case for planes on an aircraft carrier now?" Yes, but also no.
Average speed of a modern aircraft carrier (Nimitz class): 30+ knots (56+ km/h). With 85-90 aircraft carried and 5,000 crew. The major differences?
* Airplanes need maintenance while not being operated, but not fuel. Dragons will need to eat regardless. While it can be argued they'd need less calories if inactive, they still need to eat.
* Airplanes don't need to be entertained. No matter how well trained, no animal will cope well with being cooped up for long stretches.
* Airplanes don't start fights. No matter how well trained, every animal will have moods, and sometimes being playful or even spiteful could be that mood.
Adding to all this, there's a point from Starfish Prime's answer. Carrying the dragoon themselves (without a dragon) would be more effective. And you can fit those archers in far more places than you could ever fit dragons. Granted, a ship's less maneuverable than a dragon, and slower to boot. 160