I was out refereeing in the Cheshire FA Womens Cup a couple of weeks ago.
For as long as I can remember you've had to play to a finish on the day of the original tie (Three 90th minute equalisers in as many years really do piss you off) but this year a replay was indicated should the original match be drawn.
This was to take place the following Sunday, i.e. seven days after the initial match.
I'm not sure whether the seven day rule is applicable for CSC ties or not but the FA nationally set out the order of precedence of competitions which will always place a County FA competition above any League (other than the Premier League which is under the auspices of the FA nationally)
Also, I believe that entry to the Competition is mandatory for those Clubs not granted an exemption.