You should delete the routes from IE connection settings and any mods your application makes to the registry when the user clicks on the X in Fiddler. It seems like when you exit using File > Exit it does a "clean" exit. Please make this consistent.
Please be specific about what exactly you're seeing, including which registry keys are modified and what connection settings are changed. Please also provide Windows and Fiddler version information.
Fiddler should have been working properly here for well over a decade; the only known problem is that if you have Fiddler running when IE10+/Windows10 are installed, a bug in Windows means that Fiddler will not properly be detached from as the system proxy. Restarting Fiddler after login should resolve that bug.
Please be specific about what exactly you're seeing, including which registry keys are modified and what connection settings are changed. Please also provide Windows and Fiddler version information.
Fiddler should have been working properly here for well over a decade; the only known problem is that if you have Fiddler running when IE10+/Windows10 are installed, a bug in Windows means that Fiddler will not properly be detached from as the system proxy. Restarting Fiddler after login should resolve that bug.