Are sleep and dreams the key to learning?

Yes.  

“The challenges that the brain had grappled with during the daytime replayed in the mind as the subject went to sleep… (S)ubjects who spent more time dreaming… demonstrated a greater improvement in their skills…”

Via Dreamland: Adventures in the Strange Science of Sleep:

Subjects who did not get to sleep before their second shot at the puzzle showed little improvement. Those who slept eight hours, meanwhile, solved the task 17 percent faster. But that wasnât all. The subjects who figured out the hidden, easy solution to the puzzle completed each set approximately 70 percent faster than their peers…

And:

Stickgold let the subjects fall asleep normally in rooms in his sleep lab. He woke them up not long after and asked what they were dreaming about. Approximately three out of every five replied that they saw falling Tetris pieces. The challenges that the brain had grappled with during the daytime replayed in the mind as the subject went to sleep… (S)ubjects who spent more time dreaming about the game demonstrated a greater improvement in their skills the next time they played than those whose brains hadnât relived its experiences during sleep.

What if you can’t get a good night’s sleep but need to learn new info? Even naps can help:

If you canât get in a full nightâs sleep, you can still improve the ability of your brain to synthesize new information by taking a nap. In a study funded by NASA, David Dinges, a professor at the University of Pennsylvania, and a team of researchers found that letting astronauts sleep for as little as fifteen minutes markedly improved their cognitive performance, even when the nap didnât lead to an increase in alertness or the ability to pay more attention to a boring task. Researchers at the City University of New York, meanwhile, found that naps helped the brain better assess and make connections between objects.