Or you can just say 'saya kira dia perempuan'
So why does one use "dia orang perempuan" sometimes instead?
Saying "orang perempuan" is just really thorough (probably too thorough?
"dia orang perempuan" literally means 'she's a female person.' It isn't wrong, however.
Is 'bahwa' needed here, or can it be left away?
Yes, it can be left away, though it would be a bit more informal.
Why is "Saya berpikir dia bahwa wanita" being marked Incorrect???
Perempuan means "female"
Wanita means "woman"