I know of people that has brought it on the first day. but I would just have it already in your car ready and just bring it in when y'all are getting released. My hospital didn't let me carry out my son in his car seat though. after they done the car seat test they told me to go put it in the car then the nurse walked out and handed me my baby after. Every hospital is different though.
Up to you. I brought my sons the day before discharge but he was also in the nicu and nursery after and didn't get discharged until 11 days after he was born so the day before for me cause they had to do the carseat test