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.
You are more likely to get higher scores if your element matches rather than opposes the arena's element. I put a plant in fire and got bronze, plant in plant and got gold. I believe that is the only difference.
I looked it up before too because I had a similar question. I did not find much on this actually, so I tested it myself and that outcome was always mis-match = bronze or silver match = silver or gold. (I unfortunately decided to test this after all my dragons were level 7, not 10 yet but all were at least 7 so many came out silver either way).
Best Answer
Unfortunately there are no guarantees as far as dragon breeding in Dragonvale and it really comes down to luck. Rare breeding may be successful on the first try and it might just as easily take until the 50th try. You can supposedly grant yourself a slightly better chance at rare breeds by using the Epic Breeding Island. With an estimated 3% chance for the Sun and Moon, it would mean that in the average case, it could reasonably take 34 attempts, but just as some people get it on the first go, others don't until the 100th. You could have been playing this game since it came out and still not have a Rainbow dragon because it's just that rare.
To figure out your chances and get an idea of how many attempts it may reasonably take you in an average case, you can check the results that other users have submitted of their experienced by checking this handy breeding database which provides information on the success rate of particular pairings in getting a specific dragon across how many attempts. Still no guarantees though. The results here seem fairly optimistic and you may just as well halve or quarter the success rate as people are far more likely to post success than failure.
Further indicated combinations that users have claimed more successful can be found here. It also states here that "If all else fails move onto another combination. Sometimes it seems like one combo will work for one person but not another."