Black gloss will show all of the flaws in the paint work so expect to spend a lot of time sanding and featheringout imperfections using epoxy resin filler.
I would have reservations about painting a door black if it is South facing due to the amount of sun light that it will get.
My paint of choice is Dulux Trade Weathershield oilbased gloss but if the door is protected from the elements then standard trade gloss will be fine. Undercoat will be grey.
Drying times at this time of year will require leaving the door open all day.
A better option might be to use an exterior grade acid cat spray but the paint costs will be very high and you will need to get a HVLP sprayer. Touch dry will be minutes. You will be able to take the door off spray the requisite amount of coats and rehang it the a few hours later.
If you do go for acid cat then make sure that it is exterior grade. I sprayed my front door not realsiing that the spray was interior only. It looked amazing for a year and then the paint started to craze.
I was able to take the door off 10 am, sand it, fill it, resand it, spray 4 coats and rehang it by 5pm.