This depends on many things.
It depends on if there are components on the back side of the board. If so, they could fall off from too much bed heat. And it depends on what sensitive components are placed where on the board (such as anything made of certain plastics, which will melt or deform at "low" temperatures.) It may be necessary to add a metal shield to prevent the hot air from reaching a sensitive part. And you should qualify the hot-air temperature; many devices run far hotter than their setting indicates.
The best way to determine all of these and more, is to practice on scrap boards. You'll quickly learn what temperature will melt the black plastic under SMT electrolytics or DIL headers, for example.
Also consider that the exact solder alloy plays a large role in this. Higher-melt alloys (such as RoHS) are generally harder to work with, while lead-based or other low-temperature alloys make everything easier, but then contaminate the board. There are a thousand things to consider.
I could then use the hot air gun to (de)solder selected components. Or (in case of a fresh board) could I just ramp up the temperature of the preheater until the solder reflows?
Well yes, but consider that there is a large void of cooler air above the board, so you'll have to crank the bed heat fairly high to achieve this. It's fast and easy to flow some hot air from the top once the board is warm (using the hot air tool.) I'd recommend that, rather than risk de-laminating an unknown board from too much bed heat. By all means, try it on some scrap boards though. You may find that not all boards are created equal, and some boards will delaminate at a much lower temperature than others. I've noticed no discernable pattern between manufacturer/age, so it's really a try-it-and-see approach.
To cool down, should I lower the temperature using the preheater to a maximum temperature first (say from 200°C to 100°C), or can I just remove the board and let it cool down on the side?
This depends on the sensitivity of the components used. In general, it shouldn't hurt anything to ramp-down the bed temperature or even just shut it off (and let the residual temperatures decline.) But many devices do have a time limit at soldering temperature, so just don't keep the board too hot for too long. If you look at the reflow data in a typical datasheet, simply removing the board from heat (and placing in an airflow-less area like on a enclosed wire rack) can be fairly comparable for some devices/boards. Other devices may require very specific conditions. Just don't blow cold air on it (or otherwise quench it, etc.) as this can cause components to crack. Incidentally, the board might make some faint crackling sounds during cooling as everything shrinks slightly and stresses are redistributed. But if it makes a loud crack sound, inspect for something broken; failure during cooling is a remote possibility, especially for ceramic devices.
