The clothes shopping trip could be completely pointless, since he could easily become a girl of a different build and height at any time.
I know this is a comedy, but that female lead ain't great at critical thinking.
People of either sex can be great at cooking. People of either sex can suck at cooking. That is not a reliable test.
I think Nagomu is the girl. Then again, I wouldn't be surprised if both of the twins were the same sex.
Nozomu seems rather aggressive and sadistic.