Will I lose Quiet when replaying a mission

Yes, it will. The "restart mission" is exactly what it implies, even though it might not be so obvious at first (it sure as hell was not for me).

Originally I personally thought that restarting the mission was intended for re-watching cinematics and starting over, with your current actions still intact.

Personally this makes me love the game even more since now I can start over if I feel that my performance was lacking and knew I could do better.

TL;DR : Yes! Don't settle for that E-rank when you can get an S-rank!

On the iDroid map screen, pressing up or down will scroll through available commands. One of which you can select Quiet's sniping point, and another will let you select a location to scout.