"You are a woman and I am a boy."
Translation:Ty jesteś kobietą, a ja jestem chłopcem.
Yes, you need subject in second part of sentence like this. You can omit "TY" but "Ja" has to stay, so the change of subject is clear.
I translated "you are a woman" as: "pani jest kobietą" - which apparently is incorrect.
Wouldn't it be possible to use 'formal you' in a sentence like this? There seems to be a difference in age between the boy and the woman. As a native speaker of Dutch, I would use formal you ('u') in such a context. Maybe in modern Polish difference in age does not require formal you?
Your sentence is absolutely correct and most of the time you would use formal you here. One can imagine a situation where it would not be required, for example a younger brother speaking to his older sister.
Could you translate it as "Ty jesteś kobietą i ja jestem chłopcem." What is the difference in using the conjuncitons "a" and "i" ?
"a" gives contrast. "i" is the basic form of "and".
As there is obvious contrast between 'you' being a woman and 'me' being a boy, "a" is the only option.
There are some contexts where both work and mean more or less the same, but that's not very common. What comes to my mind is that you can use both after "między/pomiędzy" (between).