Generally they will breed a child of the same exact type, at least 95% of the time in my experience. However they can breed a hybrid of their elements.
When breeding two Chrome dragons (air/metal) repeatedly I found that once I got a Rust dragon (Metal+water)! This is because, apparently, on rare occasion an Air dragon breeds as a fire or water dragon instead of an air dragon.
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.
Best Answer
"Summon Dragon" (previously referred to as "Call") is a purely cosmetic behavior of the Perch of Kairos. Once you have acquired Kairos, he will either sit on the Perch or lazily fly around the area. If he is on the perch you'll see that "Summon Dragon" is not selectable. If he is flying around, pressing the "Summon Dragon" button calls him to the Perch. Within about a second he will teleport there with a time-dragony-wormhole-looking animation.
I don't believe Kairos needs to be on the Perch to use his "Activate" ability, so I see no mechanics reason to use this other than because you'd prefer Kairos stop flying around. I have seen a few account of him getting stuck, but it's unconfirmed if this ability can "unstick" him - and I highly doubt that is its core purpose.