I actually don't like any of the answers, so I chose "aliens did it".
Sexual orientation is evidently either immutable or very nearly so in adults, but it's not really clear that this implies a biological determinant. Also, the causal factors might be partially biological, partially social, or something like that. It might not have the same mechanism in each sex for that matter. Women seem to be bisexual more often than men, who more often seem to be attracted to just one sex. If I had to guess, I'd say that biology (though not purely genetic) factors probably play the largest role.
With gender identity (or at least gender-related psychological traits), there's strong evidence that there's a large biological component, but of course there's a cultural overlay that is socially constructed. It's become fashionable to insist that the entire thing is a social construct, but this seems to be mostly just an piece of dogma.
I don't know of any data that could really address the roots of transsexualism as such, though.