Doesn't it seem like the only people Walsingham shows affection to in the movie (aside from the queen) are handsome, rather young men?
Is there speculation about him in real life...or was this just a textural element added for the film?
The film depicts Walsingham as sexually ambiguous. From what I have read about him, the real Walsingham had two marriages (the first ended with his spouse's death) and a few children. I've not come across any indication that he had interests or dalliances with the same sex.
If anything, Rush's Walsingham is a combination of the real life person and, for dramatic purposes, a typical Machiavellian character, which often featured in the plays of the time period.