Can I restart X without having to close all applications in my current session?
You can't.
What you can do is start a new X-session on a different tty (screen) which should leave the apps running in the first session untouched. (This is how some display managers [e.g., KDM] allow multiple log-ins of different users using the the same physical display/keyboard.)
Source: http://www.linuxquestions.org/questions/linux-desktop-74/how-to-restart-x-without-having-to-close-connected-applications-689430/
The apps started over X will be killed if you restart X. You cannot avoid that.