I have never watched the show, so apply relevant knowledge to your interpretation; but going only on the text:
'Happily' doesn't suggest 'gay' here. It just means that Blanche would have been willing to pursue this man even if he were in a relationship if it were an 'unhappy' one. She says 'damn' because given that his relationship is 'happy' she has no excuse to pursue him (apparently the likes him alot!).
By the way Dorothy's first statement 'or what' isn't a question (you put a question mark) it is a statement. It means 'he isn't single'. 'Is he single or what' is the American way of saying 'is he single'.
'Or what' is an addition that Americans sometimes add to questions; it doesn't change the meaning. So Blanch plays on this and replies 'or what!' (a statement, not a question) meaning 'no, he isn't single'. Disappointed Blanche tries again: well if he is in a relationship, perhaps he is not happy? ('I still have a chance!') But no, he is happy. So, VERY disappointed, she curses 'damn'.