Does your car produce heat before the engine warms up? Even when it is on recirculate?

unless you have heated seats all the heat coming from your car comes from the air vents which comes comes from a heat exchanger core (basically a small radiator) in your dash that engine coolant runs through so untill that coolant heats up it's unlikely that you would get any actual heat from the vents no matter whether it's set to recirculate or set to heater or vent.

From what I can remember (when my heat actually worked) it did blow warm before the thermostat opened. The coolant lines for the heater core circulate in the engine bypassing the thermostat and warming the interior faster iirc.

that is correct but it still won't be fully warm till the coolant starts to warm.