My initial response would be no. But I suppose it would depend on the specific screen protector and/or how it was applied. Did you happen to use the device before the screen protector had been installed? How did the screen sensitivity fare then?
I use the HD contour on from bodyguardz and I have no issues with typing on the screen. Mine is just a regular plastic one. It might be different with a tempered glass one though.
No, no screen saver. I use a wallet style case so the screen is covered when it's in my bag or whatever. The screen seems pretty tough, relatively speaking.