I have no idea why they wanted Hammerfell so much though. That's another topic, but it did seem to be a primary objective for the war, didn't it?
South Hammerfell was colonized by high elves before Men, at that time they called it Hegathe. Maybe the Thalmor just make claims for every territory once inhabited by their kind. Or:
- they just want to surround Cyrodiil, both on earth and sea
- there's some old dwemer/elvish stuff they want
Back on topic, we don't have much details about the consequences. The moons are not juste idols for khaajiits, the make them the moment they are born. Bethesda never gave details but imagine that without the moons, there are no births at all: all babies are born-dead, or turn to horrible and mindless monsters the parents must put out of their misery... We can imagine that the whole event was made for the sole purpose of preventing the Mane's birth (thus leaving Elsweyr without a proper leader) but turned horribly wrong and cost the lives of all the khaajiits-to-be for two, very long years.