It's always difficult when we do not know the format of your date which you didn't advise. Here is a rename if your date format is 'day mm/dd/yyyy' - not tested.
Ren folder1 folder1_%date:~-4%%date:~4,2%%date:~7,2%
Renamed folder should be in the format folder1_yyyymmdd
You can substitute Copy for Ren when you copy the folder, no sense in copying then renaming.
Hope this helps.