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So the Dunmer refugees in Skyrim weren't really fleeing the Great War and Thalmor like the Imperial refugees were... but the Thalmor coming to power in more places in the Empire probably did not help matters, considering how much the High Elves and the Dark Elves typically hate each other.
Answering your questions here instead of in-story. Morrowind has a varied climate. The part that borders Skyrim is cold, but the province stretches fairly far south, where it is warmer. It's up to you where Sheliah is from.
As for the refugee situation, Morrowind has essentially taken the brunt of all the major events in the Elder Scrolls games. First there was the Oblivion Crisis, which was a war between the entirety of Tamriel and the Daedra (basically demons if you're not familiar with them.) Every province was affected by it. Then there was the Red Year, where the moon literally fell from the sky (and onto Morrowind, of course) causing all manner of natural disasters. And then after the Red Year, Morrowind was invaded by the Argonians of Black Marsh, the province bordering it to the south. It was easy for them to take advantage of Morrowind's weakened state and the fact that the Empire pulled most of its soldiers out of there during the Oblivion Crisis to fight the Daedra.