Is vapor of mercury conductor or insulator?

Question

Liquid mercury is definitely conductive.

It is also possible to 'convert' liquid mercury
into gaseous phase without any phase
transitions, just turning over its critical point.

Answer 1

Mercury vapor is conductive, but in an unusual way, see the first reference. The mercury ionizes at the cathode, but physically carries its charge to the anode. It's not classical electrical conduction.

In a mercury vapor lamp, the mercury facilitates a gas-discharge arc, somewhat closer to conduction (third reference)

I don't understand your second question. The transition of a material from liquid to gas phase is by definition a phase transition.

Answer 2

I think you sort of got confused in order to convert liquid mercury into gaseous mercury you must take it through a phase transition. Usually vapors are not conductors for a variety of reasons.
Metals are such powerful conductors because in solid and liquid for they share electrons in their valence shells very easily allowing charge to flow with much resistance. However as a gas the atoms would need to be in remarkable proximity to one and other in order to share electrons.
If you kept the pressure high enough it should be possible for this sharing to occur however you would need to maintain remarkably high temperature in order to keep the mercury gaseous and you would be stretching the bounds of the definition of liquid and gas