Some people who dress in the clothing of the opposite sex are called transvestites. The vast majority of transvestites are heterosexual, not lesbian or gay men. No, lesbians are lesbians because they form loving relationships with women. Gay men are gay men because they form loving relationships with men. Neither form these relationships because they hate the other gender.
Many lesbians have close male friends, and many gay men have close women friends.
No, lesbians and gay men do not want to be the opposite sex. Within the gay community, there are many who have challenged and discarded stereotypical sex roles but this does not mean that they are trying to be the other gender. Gayness involves celebration and affirmation of oneUs gender, not a rejection of it. Some people confuse gayness with transsexuality. A transsexual is a person who has an operation to become the other gender.