Could a small Cessna Aircraft take off from an aircraft carrier?

Possibly. B-25's did it.. see doolittle raid..

Easily into the wind. "from the far right of an aircraft" What the heck does this mean??? "could the total length of the carrier take off' The carrier would never take off.

Huh? Of course it matters whether the carrier runway is banked to the wind or against the wind, and how strong the wind is. I saw a video recently of a small plane landing on the top of a moving van that was going about 40 mph.

Piece of cake. It can do it with it's eyes closed.

Since a modern aircraft carrier maintains at least 25 knots of speed during flight operations, you are guaranteed at least that much of a head wind. A 172 with that much head wind could take off from a carrier easily - you would only have to accelerate another 35 knots.

No problem taking off or landing. There may be a slight problem getting permission to do it though.

Yes, it's perfectly possible, particularly with the carrier turned into the wind (as it usually is during launch operations), which can easily produce a 40-knot wind across the deck or better. But even with no wind, a Cessna can take off. There are no obstacles to clear at the end of the deck and the aircraft will already be 70 feet above the water after clearing the deck, and the deck is nearly 1000 feet long. There are a lot of variables (the temperature should not be too high, and it's best if the aircraft is lightly loaded), and there are some conditions in which it would be difficult, but overall it's possible.

You might be able to -- BUT I WON'T DO IT!

Yes. No modifications needed. The windspeed plus the speed of the carrier through the water gives the aircraft a considerable headwind, and shortens the required takeoff distance. And you don't have to be at the maximum allowed gross weight.