To each there own. Although I don't understand how someone can believe they are biologically the wrong gender, I have no problem with it as long as it doesn't affect me. I am not sure how I'd feel if one of my kids, grandchildren or great-grandchildren decided they were transgender. It would definitely take some getting use to.
As for beauty contests and other traditionally gender specific competitive events, my personal feeling is transgender folks should compete against one another and not against biological males or females whichever the case may be. If were one of the female contestants in the Miss Nevada competitions, I'd be pretty upset if a transgender person was the winner. No matter how beautiful that person might be, they will never be a biological female.