I am certainly no expert, and the body of water will determine many of these variables, so there is definitely no one answer...but that said, here's my thoughts.
How much wind - In my book, at least 10kt minimum. I can paddle consistently at 6-7mph so to get any push from the wind, 10kt seems like a decent minimum. Fun conditions begin around 15-20kt and things get pretty damn wild at 25+
Does the wind need to be present for some time prior, or are the wave effects immediate? Depends a lot on the size of the body of water, but in general there is a pretty fast response. A larger body requires more time to build, but I'd say within a few minutes of a strong wind picking up you will see appreciable waves building. On a small lake, the effect can be almost immediate (like less than a couple minutes)
How deep does the water need to be? Dont know specifically but I have downwinded in depths between 15 and 150 ft of water. Shallower water will build larger waves at lower windspeed.
Is the contour of the bottom relevant? Yes. Surfline.com has a great info archive on bathemetry (the science of waves interacting with the floor). They mostly focus on long period ocean swell for surfking, but the same principles apply to short interval waves as well. I recommend a good read of their info. It helped me understand wave science a lot more.
Do current or tides need to be consistent with direction of wind? same, or even opposite direction - Tide direction / current direction will change the sape of the wave depending on if the current/tide is going with or against the waves. If going with the waves (like an incoming tide) the wave gets lengthened out to some degree because the water is pushing the same direction as the air.
When the current runs against the wind, it has the effect of steepening the waves because the water is pushing the bottom of the wave against the top creating a steeper face.
The most easily surfable waves (IMO) are when there is a strong tail wind pushing you into the current. These are the conditions at The Gorge in oregon where the north american downwind chaps are held (for this exact reason, among others)
Does it require a very large body of water (100's or 1,000's miles), or will a small bay or lake work (~3-5 miles across)? The size of the water will determine the maximum wave height, wave interval, and how quickly the waves build. You can certianly get surfable waves on small-medium lakes. I belive paddle pop or sski lives on a lake in scandanavia and has posted video of very surfable waves on a medium size lake. Generally though, a larger body of water is better, as the maximum wave size potential is higher. This is why you have 50ft waves on an ocean and 5" waves on a pond.
Any other requirements - Staying upright? swimming makes you go slow. The better the downwind, the less a skinny boat matters. for this reason I am going to try and get 2 skis, a swordfish for small/medium days and a bluefin for the big stuff.
What makes one "run" better than another, IF they have same wind speed? Other than logistics, clear path, scenery, rocks, sharks...
- The best runs I have ever had, had 2 things in common - an interval of 7-9 seconds and a wave height of 3-6ft. The interval determines wave speed. Per Surfline's Bathemetry information, wave speed = 1.5*Interval, so a 8 second wave moves at 12mph, the high end of what most intermediate paddlers could catch. Any longer interval than that and it takes super human strength to get up enough speed to catch and stay on the wave, so the long period ocean swell that surfers love is pretty worthless to a ski (although with enough wind the wind waves can push you onto the long period swell and give you short slingshots at very high speed, my record being 13.6mph down the face of a big swell on a downwind). Conversely any shorter than 5 seconds and the wave is moving too slow to be overly useful.
Likewise, the ratio between height and interval determines wave steepness. The steepness of a wave plays a large part in how surfable it is (steeper being better, until the point you bury the nose or pitchpole).
I think the most important factor is the wave interval followed by height. these are both tied to wind speed and body of water size/depth.
If I could choose my conditions at will, here on the southern california coast I would choose 1.5m height at 7 second interval with a 15-20kt tailwind. That's perfect in my book.
Other thoughts?