"Are you writing?"
Pisze Pan(i)? or Czy Pan(i) pisze?. Without the subject it doesn't make sense.
It makes sense, but it sounds rude.
It's for when you don't want to show any familiarity implied by the second person forms, but you don't want to show any respect either.
I don't know, maybe it's regional (Are You from the eastern Poland or something?). For me, the third person is for sure always the third person (I mean some other person, not included in the conversation), unless you use a title (pan, ksiądz, profesor, Wasza Wysokość). For example, telling someone niech usiądzie instead of niech Pan(i) usiądzie doesn't make any sense at all.
You can sometimes encounter the third person singular without a title (but directed at a person taking part in the conversation), but this sounds extremely rude (and I'm not claiming it is grammatically correct). An example from the movie "Dzień świra": "Opuści, zsunie spodnie. [...] Wypnie. [...] Niech nie kuli jak pies przy kupci." (https://www.youtube.com/watch?v=NQOq7g1FQ84).