When does the document verification round starts in JAC Chandigarh counselling?

Yes. You can. Both counselling have different purposes. JoSAA is for admission to IIT and NIT+ system, while JAC Chandigarh is for admission to  Dr S. Bhatnagar University Institute of Chemical Engineering and Technology; Panjab University, Chandigarh (Formerly Department of Chemical Engineering & Technology); University Institute of Engineering and Technology, Panjab University, Chandigarh; University Institute of Engineering and Technology, Panjab University; Swami Sarvanand Giri Regional Centre (PUSSGRC), Hoshiarpur; Chandigarh College of Engineering and Technology (Degree Wing), Sector 26, Chandigarh; and Chandigarh College of Architecture, Sector 12, Chandigarh.

Admission via both JoSAA and JAC Chandigarh is based on JEE Main ranks.

Yes, it is possible to get admission to Panjab University with the rank of 500000 in JAC Chandigarh. Apart from other exams, Panjab University also accepts JAC Chandigarh for various courses. It is difficult to get admission to BArch at this rank; but getting admission to BE is possible. At 500000 rank in JAC Chandigarh, it is very much possible to get even the most difficult specialisation CSE, at various affiliated colleges such as University Institute of Engineering and Technology, Panjab University and Chandigarh College of Engineering and Technology, for the General AI category. Checkout this page for more details on Panjab University cutoff.