Are opinions of 'women are weak' or 'women can't be rulers' acceptable?
No more nor less acceptable than the other players make them.
I know characters can be mustache-twirling-kitten-eating-bad-guys who have offensive opinions AND that does not mean that the players are evil schmucks too. But, in the culture of BM, is it cool to denigrate women solely based on their gender?
I don't think it's ever "cool" to do so. But while BM's flavour of "medieval" doesn't generally include the kind of misogyny that characterized the actual period, I see no reason why a given individual character in the game can't express such beliefs.
There's also no reason why any other character in his realm has to treat him as anything other than a dangerous lunatic.
