Paint sprayer suitable for both large areas (exterior) and finish work (cabinets)

First off, I'm no expert on compressor/air type sprayers. Most of the time, these type sprayers are used for very thin paints such as automotive, lacquers etc. They also have fairly small paint reservoirs attached to the gun itself. They tend to put out a lot of over spray and should be used in a controlled area like a spray booth with proper ventilation. I never see this type of sprayer on building jobs.

The airless sprayer, such as the one you noted, use a high pressure pump to push paint through the hose to the gun. They deliver a good pattern (depending on tip selected), are easily controlled and since they don't mix air with the product, have far less atomazation of paint in the air. the other huge advantage is that the pick-up hose can be put directly into your 1 or 5 gallon can of paint. Follow the directions on priming and cleaning, but it is easy. I personally have a Graco model 17 and love it. It is a professional grade sprayer and retails around $450-$500. Just two weeks ago, I and a helper painted and back-rolled a complete two story house and detached garage in one day.(very long day, however!) This saved 3 or 4 extra days of labor and the results were fantastic. We shot both semi-trans stain on the house and matching color solid stain on the garage. This sprayer handled both products easily. (I always use apx 10 to 15% flowtrol with thicker solid latex based products)

The only disadvantage of this type of sprayer is on very small jobs. It takes about a pint or more of paint to fill a 25 foot hose and closer to 1/2 gallon for a 50 ft hose when priming the unit. This amount product is usually wasted at the end of the job when you get ready to clean the pump.

From the type of jobs you plan to do, I would recommend the airless sprayer hands down.

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.