The best type of mower for the very first mowing is a sharp bladed rotary mower. These types of mower perform well when the grass is a bit longer - the situation we have when the first mowing is needed. Obviously you cannot mow the lawn if the turves have not rooted sufficiently to be held in place, otherwise the turf may be sucked up and chopped up by the mower.

The first mow can be done when the turf has rooted to your soil.

Mowing can be done at any time of the year, so long as there is not a risk to frost/snow as this can "whiten" the lawn.