"Oh dear, this is my husband."
Translation:Obh obh, seo an duine agam.
duine = a man, so "seo duine agam" would be "this is A husband of mine/at me", as if the speaker had more than one; therefore, assuming that the person only has one, is has to be "this is THE husband of mine/at me". Not sure about your second question, though - I'm tending to equate seo and sin with voici and voilà respectively, i.e. including the verb, as it were. (I think you've also done French??)
According to this page http://www.akerbeltz.org/index.php?title=Possessives_and_syllabic_structure_or_Ar_n-Athair_a_tha_air_n%C3%A8amh "duine" is alienable and you don't say "mo dhuine" (although you do say "mo bhean" and "mo chèile", not "a' bhean / an cèile agam").