When 3D-printing on an 20*20cm, I've heard that the quality of the printings get worse if I fill out the board... Is it true? Should I keep it to small amounts at the time or doesn't it matter?
No, that's not (entirely) true. There might be some loss of quality if you print multiple objects at once, because when the printhead "hops" from one object to another it might leave a mark or ooze out some material. Also, a large number of retractions in a short period of time might lead to inconsistent extrusion.
However, none of this is particular to "filling out the board" as it happens even if you print only two objects at a time (or even when you're printing only one object with multiple islands).
It all depends on your printer (and in particular how well it handles retractions). If you're willing to do a small amount of cleanup afterwards (to remove the strings and blobs) then printing multiple objects at a time is completely viable.
This would depend on what type of printer you have and to some extent, the design of the printer.
Cartesian (i3 style) - Should not have any issues printing a full bed. I have seen designs and variants of printers where the designer did not make the belts parallel to the travel axis (placing the mounting point on the carriage higher/lower than where it come off the pulley/idler) so as you move closer to the pulley/idler the angle starts to change dramatically. While not extreme, this will introduce error in that axis.
H-Bot/CoreXY - Should not have any issues with a full bed as nothing is changing depending on hotend location. I know there has been some concern about keeping the gantry square on H-Bot designs due to the way the motors pull when moving in certain directions but I believe that applies to the entire print area. Both designs will have issues if belt travel is not parallel to axis travel as stated in the cartesian section.
Delta (Kossel/Rostock) - Have a varying axis resolution depending on the location of the hotend. As the hotend gets close to an individual tower, the carriage on that tower has to move very little to move the hotend which depending on your pulley selection could affect your print resolution. The sweet spot for a delta in terms of speed/resolution is the center of the bed. The center is not the highest resolution but is the largest area where the resolution is relatively stable. I've tried to summarize all that I could from here, but I've included the link if you wanted to read more about it or see the graphs.
Finally, as Tom covered. All three above will have issues with the entire bed depending on your hotend calibration and how well your printer was built. Delta's are fussy about build quality but you will solve most issues on a Delta if you level the bed before using any kind of bed leveling/probing.
Just to add to the other answers already here, I've also had problems with layer adhesion when filling up the print bed on an extrusion printer. The longer the print head spends working on details around a single layer, the more chance the layer has to cool off before the next layer gets started. I've started to suspect that layers that take longer to print end up with less adhesion to the next layer.