Welcome in, Rilarien. As far as I know, horses were not used in any way by the Galadhrim; as for elsewhere in Middle-Earth, as in our own earth, they would not in fact have lived in all areas -- although they were, as you say, in widepsread use across the continent, they were largely under the ownership of people.
Rohan, of course, supplied much of the horse stock of the Western Lands. I suppose that since we are talking about a large area, and one which does largely consist of plains and expansive, flat country (the majors of this being Eriador, Rohan and Wilderland, Gondor and the South, and Rhun to the Eastward), horses were needed to cover the miles.
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