An DC offset, if not too large would mainly change the On/off ratio. So it would not be a big problem. The bigger problem is more like higher harmonics and their variations, as they can move the zero crossing. Something lime switched mode power supplies can be such nonlinear loads that can cause such waveform distortions.
For a longer time to measure the frequency there are different ways of "averaging". Depending on the type of noise, different methods to calculate the frequency are best. With noise due to white phase noise, like jitter from a poor trigger, a good way would be a kind of linear interpolation of zero crossing time versus count number. If the trigger is perfect and noise is more like a poor oscillator the better method would be just looking at the 1 st and last time in the interval.
With the mains frequency it's more like the trigger noise case, though there are true variations in the frequency too.