Z-axis comes down and presses the bed while printing
When the head/nozzle assembly runs into the bed, you need to watch out for:


Check for Endstop problems:


Is the switch properly wired up?
Is the switch working?
Is the switch mounted firmly to the frame and correctly adjusted?

Level the bed as level as possible; what is meant here is that the bed needs to be parallel to the X-axis, not bubble level. This ensures that the nozzle is exactly at the same distance away at every point on the build platform. You achieve this by setting the Z endstop as such that the nozzle is close to build platform with platform screws screwed in for some distance. When the endstop is firmly attached, home the printer through the menu of the printer, try adjusting the screws such that the level is everywhere the same (in between changing the location of the nozzle carriage to set the level, home the machine once in a while). Use a piece of paper for the correct distance; a slight resistance felt when dragging the paper is sufficient. Please note that it makes a difference when you level the bed cold or hot, hot is preferred.
Slicer settings; Be sure that your slicer doesn't introduce a Z offset.




EDIT:
With respect to the additional information, you really need to check the level of the bed, level with a hot bed and a hot nozzle (e.g. PLA temperatures of respectively 60 &deg;C/190 &deg;C). Furthermore, be sure that you get enough adhesion. Use (sanded and cleaned) tape, hairspray, glue stick or whatever trick you can find to get the filament to stick well enough that it won't be picked up by the nozzle and dragged around. For me personally printing directly on the aluminium bed with 3DLAC (sort of super strong hairspray) or on glass works best, I'm not a fan of tape.