G'day Maew, i agree that the fruit is splitting because of inconsistent watering. You will find that if you mulch the
tree heavily out past the dripline after a good soaking and a feed you'll have a better water retension in the soil and it'll handle a dry patch a lot better. Citrus have a root system that feeds close to the surface which is why they respond badly to dry periods and why the mulching will help.
If this is your
trees' first year in the ground you're better off to remove the fruit at least for the first year. It's a big job for a little
tree to put out fruit so early and you'll get a better
tree in the long term if you allow the root system to develop and put its energy into establishing a good strong
plant.
The bleeding sap sounds like foot rot. I've pulled some info off the net and pasted it below. If you punch "citrus bleeding sap" or similar into your search engine there's lots of info available.
"Closely inspect the trunks and lower limbs of each
tree. Carefully
cut away any bark that's loose, cracked. or bleeding sap. Don't remove
healthy bark! And no wood need be removed unless it's rotting.
Immediately treat such surgical wounds with a smear of Bordeaux Mixture
diluted with water to a thin paint consistency. "
( Maew i think a copper based fungicide mixture would work as well here.)
Hope this helps.