I'm not certain whether Provado is systemic or not - check the bottle, and if it says it's systemic, don't eat the lemons. There should be some guidance on the label about using the product on edible crops anyway. Can't help as to why there's a variation in size of fruit, except that's fairly common, and as for the split in the fruit, I don't know the answer to that either, except to say that I know this occurs in tomatoes with uneven watering.

We are only growing our first lemons too this year, with two large fruits, one of which is yellow and slowly ripening.
I would say that the splitting is to do with irregular watering. Although you must be careful with citrus not to overpot, I would definitely move yours into a rather larger pot (say 10 ins) in the growing season, with some fresh citrus compost.
I guess you are feeding with special citrus feed and only using rainwater, if your tap water is hard.
The only disease you may need to spray it for is mealy bug which is a real pain on citrus, though there are organic alternatives for that.

Tanks Bertiefox.