Which dragons can live in the Gemstone Island

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.

Dragons keep their levels, they are only visually reduced to babies. They will also level normally after being made into babies. They will never return to adult form unless you bring them back to the Fountain of Youth again.

Dragons that could breed before being returned to baby form can breed after they're baby-ified as well, so you can breed two level 4+ "babies" after using the fountain of youth.

Dragons can be made level 10+ through use of the Elemental Shrines which you start unlocking at level 18. it's entirely unrelated to the fountain of youth.

The dragon must be at least level 10 to be "baby-fied". The dragon can not be baby-fied until it has reached level 10 otherwise it is completely impossible to turn a dragon back into a youth. The dragon will looks as if it is in it's juvenile stage at 5 hours into the 12 hour process.