Remove oil-based paint from furniture with paint stripper and pressure washer

Normally you can use latex over oil based products (never the other way around) if they are fully cured and dried. If the putty you are referring to is glazing compound, then you may have to wait a bit longer for it to cure. Glazing is very slow to cure and take paint. If you can still very easily dent it with your fingertip, wait, the solvents have not yet evaporated. You certainly can test a small area and see how it goes. However, don't be surprised if you get some peeling in a few months. Oil based paints are hard to find now, but would work better. I don't know what product you used, but there may be some recommendations on the can about painting.

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.