The physiology of fun?

Generally in a wild population, "play" as young animals provides for the development of skills that are useful in adulthood, for instance cats that chase after everything as kittens will be better chasing after stuff like mice and rodents as adults. So once an animal has learned the skill, the only practice that is needed is the actual use of the skill.

Developmentally and psychologically, there is a condition called neoteny. This is where juvenile traits carry on into adulthood. This is seen in many cases of domestication and can be seen vary well in the domestication of silver foxes (http://en.wikipedia.org/wiki/Domesticated_silver_fox). Basically, some animals don't grow up but still become sexual viable mates, and sometimes this is evolutionarily better for survival. An extreme example of this is some salamanders that never transition from their larval stage to full grown adults (http://www.smithsonianmag.com/science-nature/the-salamanders-that-refuse-to-grow-up-64827289/?no-ist)

Many domesticated animals show some signs of neoteny, for instance in the foxes in order to fully domesticate them over a short span of time, neoteny traits were selected for.

Does it correlate with intelligence? Crows are clever. Cetaceans, too, and they are known for play. If play is useful in childhood to facilitate learning, individuals who play as adults should be selected for to the extent that continued learning in adulthood is useful to reproductive success. We know it's fun for children to play. The physiology of its funitude may be co-opted into adulthood, much in the way the physiology of parental bonds has been co-opted into non-related pair bonding.

Do adult crows and otters play just as much as young crows and otters? Or does it drop off, but not as much as in other species?

Try the book "Inside Jokes" by Hurley, the Dennett and Adams. It is human-centric, but their theory of humor might be useful to you.