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.
Per the wiki:
The Sakura Dragon can be bred by using a combination of dragons whose elements include plant, earth, and fire. The in game splash message indicated that you must be level 11+ to breed this dragon.
You may have bred a Sakura dragon if you get a breed time of 10 hours, but you will have to wait till you see the egg to be sure, since it could be a Lava dragon as well. The egg is brown egg with three cherry blossoms on it, like this:
The sakura dragon is only breedable until May 3rd.
Best Answer
This sounds like a pretty straightforward bug, though not one I've heard of. You can use his "Summon Dragon" ability as DKnight points out - however that is only really relevant for him flying around. If you are having problems using his ability, I would consider that more serious.
As others point out, on iOS your DragonVale account is tied to your GameCenter login, so you can safely delete the app and re-install (just read the prompts carefully when doing so, to retain data). I presume the same option exists for Android devices, since most of your park is stored "in the cloud" - but I'm not an expert on Android. If you delete and reinstall your app, it will likely "reset" your park to a known good state, hopefully with a fixed Kairos.
Finally, check frequently for app updates. This app gets updated all the time, and may of those updates include bug fixes. Possibly this bug has been addressed in newer versions. I know they had a lot of problems with Kairos initially.