I wouldn't recommend diving the cenotes unless you had some experience diving.

I have no doubt that places will let you do dive training without being able to swim. But reputable places will make you do a swim test first. When I did my test I had to be able swim/traverse 400m. I also had to be able tread water for 2 minutes with my hands in the air.

I don't believe there is a weight limit. And prices will vary from place to place.

I recommend that you do some research at your local dive shops. Many dive shops (including overseas places) will allow you to do the theory component on line and then the practical later on. Normally over a couple of days.

I hope this helps.