Does dragon maturity (past juvenile) have any effect on breeding

Yes, the breeding time is always the same as hatching time for all dragons, rare dragons included. This means you can check your breeding time to know what dragon you're going to get; you're getting a Tree Dragon in this case.

You can verify the hatch/breeding time for any dragon with the Wiki's Dragon Reference Chart to check which kind of dragon you might be expecting. Alternately you can just look in the store in-game. Look for dragons matching some/all of the elements of the Parents (combined) that have the same hatch time as your remaining Breeding time.

Note that the upgraded breeding cave/island reduce your breeding time by 20%, but the incubation time will remain the same.

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.