What type of wood for moist exterior location that is also paintable

Do I need to sand or is a good wash down enough? Or maybe a liquid sander?

A good wash, followed by a thorough drying, followed by a light sanding with fine grit sand paper to rough up the existing surface. Sanding ensures the new paint will get a good grip on the old paint.

Do I need to prime, or is the old paint (cleaned a bit) good enough?

No, its been sealed long ago - just clean and sand.

Is there any special paint, other than exterior trim (probably satin or semigloss) to consider?

Since there's already latex on it, exterior latex should be fine.

Is there any additive I should use with the paint for flow, finish, hardness?

No.

Do I really need more than 2 coats?

If it were white over black- you might need 3. Black over white, you might only need one good thick coat.

Is there something else I am failing to ask/think about?

Get saw-horses. Lay door flat on saw horses. Remove door from hinges, remove all hardware, tape all glass. Paint ONE SIDE only at a time, let try completely, then paint other side.

You call it a patio, so I assume it is not a wood deck (which would be easier to add a railing). I'd avoid attaching it to the house though it seems that the other end of the patio roof may be tied to the house already. Consider driving a super spike into the ground at each end to hold a post that will be the ends of the railing. At a 10 foot span, you'll want one in the center of those two as well. If driving the spike just off the patio won't work, then remove a patio stone and drive the spike where you need it. You can break/ cut the patio stone to fit around the post once its in place.