If I configure this app as a ClickOnce application, once it installs from the url or shared drive, the user will not need to do anything else, correct? Or do they have to visit that url every time they boot up to reinstall/run the client?
Well if they need to reinstall it, they would have to download the setup.exe file again, but why would users need to do that? CilckOnce supports automatic updates. Visiting url is certainly not needed for running the program.
what is the best way to store user passed credentials on their system for a "remember me"?
Probably storing them as application settings. Haven't used visual studio 2012, but in visual studio 2010 you have to right-click on project in solutions explorer, go to project propeties and then create variables in Settings tab. Then you can access those variables in code using My.Settings.variableName. Not sure about security though, if you need any.
As for privileges, I think you just need a privilege to install a program. Maybe you should publish a primitive clickOnce application and experiment with it and that will answer all your other questions about clickOnce.