I concur that there is a good chance of evaporation between placing the drop and finally covering it with the lid. Once you place the drop(s) they quickly spread out and being as hot as they are they may also evaporate some water. And it does not take much evaporation to create a significant error.
If you "hover" the cover plate over the sample before closing it, you'll see condensation form. So - at least to me - it seems clear that evaporation is occurring. The only question is how much it will skew the readings. Like nyakavt, I've found the discrepancy to be large enough that it has to be managed.
Wait 30 seconds after placing the sample and closing the cover plate to allow the sample to come to ambient temp (i.e., same temp as the refractometer). This minimizes/eliminates the impact of sample temperature as the refractive index of a sample is a function of temperature.
I would disagree. You want to close the plate as quickly as possible. In addition to any benefits from minimizing evaporation, more mass in contact with the sample can only help in reducing its temperature.