I was curious on this question and was hoping someone who knows that part of history could answer. I do read Shakespeare so can confirm the first part of your question.

What I do know is that in Shakespeare's famous Sonnets, half of them were gay love poems directed at another man. About a third of the poems expressed love towards a woman. In some poems, there was a love triangle between different people. So, those Sonnets were absolutely bisexual in nature. The poems were publicaly available and had London been very intolerant towards gays, Shakespeare would have gotten into big trouble just based on those poems alone. Obviously nobody has direct proof of exactly what Shakespeare did in bed but those poems are well beyond circumstantial evidence that he put a ton of thought into gay love.

To me the question is how much tolerance existed and whether Shakespeare was at risk with those poems. Secondly a question of how open people could be openly gay. Hopefully someone else can answer that part of your question.