What value of conversion gain is achieved ?

Check transistor operation point as well as local signal amplitude for gilbert quad MOS-FET.

Yes, you want to keep everything in saturation, but a better way of thinking about the mixer is that when the switching core is fully switched it looks like a cascode amplifier. You want to make sure the cascode amplifier is in saturation and that you keep the switch transition as quick as possible. There can be substantial nonlinearity and noise during the switch transition, so keeping the transition short will minimize the effect. However, the signal swings at the output should not be large enough to cause the switching devices to go into triode (or else there will be substantial nonlinearity).

The other source of nonlinearity is the input differential pair itself. If the output swings are not causing the switches to go into triode then check to make sure the swing at the input of the differential pair is not so high as to cause the input pair to generate nonlinearity.