Make sure you go for a short nap roller. Something like the UNi-PRO 100 x 4mm Nap Little Ripper Microfibre Mini Roller Set would be perfect, and it will give you a very smooth finish and an excellent result.
Painting all surfaces of your door, including the top and bottom, is highly recommended. Any unpainted section will suck moisture into the door and cause it to delaminate the skin, and any MDF components will expand. This is most often seen on exterior and bathroom doors where the underside is left unpainted. Raw timber is like a big sponge and sucks the moisture out of the air.
There is no reason why a door needs to breathe. There are vent holes on the bottom and top of a door, but these are not there to allow the door to breathe. They are part of the manufacturing process and allow air to escape as the skins are compressed and glued onto the frames.
It would be best if you considered painting the bottom and tops of the doors to comply with warranty conditions, and I would highly recommend doing so on bathroom, laundry and exterior doors. - MitchellMc
You have 2 options currently on the market regarding door paint: water-based paint and oil-based paint.
Water-based paints don't go yellow over time but it can be quite tricky to work with as the paint dries too quickly and as a consequence leaves brush marks on the surface. On the other hand, oil-based paints go slightly yellow over time, especially if the doors are in dark areas out of reach from the sunlight (like a basement or garage). But the advantage of oil-based paints is that they flow much easier and they also settle in a uniform way across the surface if applied properly.
I would recommend using a 4mm nap roller for both types of paints. If working with oil-based paints, add a paint extender (like Penetrol) so when you are rolling or brushing the paint doesn't dry too fast, allowing you to work at your own pace and producing a brush-mark free finish. - paintmyproperty
I hate painting doors inside. Lotsa fiddly bits with people moving about. So I made a gig to hold the door vertically outside. It was also great for installing new handles. The only problem is those little black insects that seem to come from nowhere and attach themselves to the wet paint. - Noyade
I've been told to lightly sand the outside edge of the doors, put painters tape over the hinges, put one coat of primer and two coats of semi gloss on the doors, I think I'd like to use a brush rather than a roller. - lindad
Use a quality brush to do the cut features on the door and a gloss paint roller to make the job easier and get a uniform finish all around.
I often use saw horses to keep the door off the floor, but if your doors are installed then just possibly a good quality drop sheet under the door so that you will not get paint on the floor. I suggest planning your progress to which door gets painted first, second, third enough time for the paint to dry then the second coat.
Remember to have enough ventilation in the house to facilitate drying and getting a superior finish. Remember that paint does take longer to dry in cold weather. - redracer01