Not so. A forward biased diode has nonlinearities of all orders due to the exponential. And the slowness isn't too relevant since the only thing leaving the die is the rectified part. The RF swing across the diode will be reduced by the capacitance, of course.
Yup.
Well, there's the well known IF noise blanker approach, beloved of QST fans since probably the early '60s. You just turn off the signal till the pulse is over. Another approach is a boxcar integrator with just the right sampling interval.
Cheers
Phil Hobbs