Rare dragons are certainly less likely to be bred when they're an option. It's all a matter of chance, though there's been some unproven theories that the dragon you put on the left side of the breeding menu determines the order of elements of the egg. It doesn't seem to hold true.
Generally assume the breed will yield a random possible combination, and note that rare dragons are very rare, I'd say maybe less than a 10% chance. I don't think there's any formal numbers. From experience only Rare dragons have a different breeding chance, everything else, assume an even chance of all other possible results. If there is a difference, it's not large enough to fuss over.
Special Event dragons also tend to be "Rare" at approximately the same rate as the Sun/Moon/Rainbow dragons. Generally you'll want to stop everything and start trying to breed time-limited dragons immediately until you get one, or two if you want a breeding pair.
Note the Epic Breeding Island advertises that you have a higher chance of rares. It doesn't seem extremely significant; rare dragons are still rare on the breeding island. In fact I haven't noticed a perceptible difference, but if you're trying to breed for rares you might as well try the island first.
Also note that breeding two dragons always gives you that same dragon (best I can tell), so breeding two Rare Rainbow dragons has a 100% chance of giving a rainbow dragon.
I believe there is a trophy for having all dragons at once, and they are more likely to make the breeding combo you want. Other than that, they are there for whatever tiny amount of money they give.
Best Answer
You need Shrines to level dragons past level 10. Shrines allow dragons to level to level 15 after one condition is met, and then allow a max level of 20 when a second condition is met.
Once you have a shrine, you'll see a counter counting how many level 10 and level 15 dragons of the matching element you have. Once you hit 50 level 10/15 dragons for that type, you're permanently allowed to upgrade dragons of that type to level 15/20. All dragons you've ever had count toward the total; if you sell level 10 dragons they still count toward this total, even if you had them before the Shrine or even if you sell the Shrine.
Shrines only work for dragons which have their specific element; a hybrid Earth Fire dragon is covered by both the Fire shrine and the Earth shrine, and it's level counts towards progress on both shrines.
The Dragonvale Wiki has a list of Shrines and their costs/level requirements, but you start unlocking them at level 18 with the Plant shrine and unlock one element;s shrine every level after that, until level 25. They start at 500,000 and increase in cost by 500,000 cash each new element.
Fun Fact, the shrines change color when you meet goals; they start Bronze, turn Silver once you have 50 level 10 dragons, and turn Gold once you get 50 level 15 dragons.