Do you mind...?

I will respect his/her choice. Personally, I don't care about her sexuality as long as it's not me that show in her mind. Beside, being a friend is about accepting who they are and support their choice. You can give advice but it's up to them to listen or not.

it wouldn't brother me at all but i'm lesbian lol
it wouldn't stop be from being friends with someone. sometimes its really hard to tell people because you don't know how and you don't want to lose them so you keep it from them. maybe give them a way to tell you or bring it up

I would be understanding. Coming out of the closest is a scary thing and it takes some people time to build up the courage to confide in their family and friends.

I would definitely respect his/her privacy and I'd also let my friend know that I'm here if he/she needs to talk. It's always nice to have someone you can talk to, right?
I would never stop being a friend to someone because of their sexual orientation. It doesn't change who the person is.

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there is nothing wrong with it as long as he/she does not invade your privacy as well... If you are not open minded I guess its time to let go of your friendship which can be sad

I would surely respect his or her feelings. For me the sexuality does not matter in anyway. And if my best friend had not told me about it before I may ask why he/she didn't tell me. I expect the reason can be that somebody can be uncomfortable telling it in the beginning times of a friendship so I would understand that as everybody of us has some reservations about certain things and we would not like to bring up the topic upfront. So okay for me if some one has a genuine reason for not stating something like this at the beginning. But if that is not the case and it was hidden purposefully that I would consider it to be a breach of trust and may rethink my relation with the concerned individual.