Is it enough if I just clean the screen with a microfiber cloth or does it have to be a special agent?
Depends on the level of soiling, it doesn't get dust away from stains.
Sufficient, only clean with a damp cloth and only with water.
Unless you somehow spill coke on the monitor.
Yes, I'll use windscreen cleaner. Under no circumstances should you use gasoline or solvents.
A microfiber cloth is great for dust (light soiling). Otherwise a mild window cleaner if there are real stains on it or a display / monitor cleaner. Although they are a bit more expensive, but not necessarily better.