I've been giving DD cheese since she took to finger foods, about 7 months. She has thumb size chunks of full fat cheese and cheese slices. I recently tried a cheese stick too, it went down really well actually. In the beginning, I watched DD closely as some-times she would bite a bit to big of a chunk off. It does seem to dissolve / melt in her mouth though and she has never gagged on it. As far as allergies, I think its mostly dairy in general.
DD is 3yr 8 months - DS is 6 months