Well, I had an interesting evening last night. First I stumbled upon this post (http://3dprintboard.com/showthread.p...inter-Extruder), where the video describes exactly the symptoms I was having with my prints. So I immediately decided to calibrate my extruder using that video and gmay3's post a reference.

I quickly found out that my extruder was a bit uderextruding (94mm instead of 100mm requested), and I used LCD menu to adjust E-steps. I decided to use the same G-code I printed before – a 24mm 20% infill cube. The result, well, dreadful.

Original settings (mesh at the top, not watertight, but very neat print otherwise):
IMG_8008.jpg

After calibration (a mess all around):
IMG_8011.jpg

I then reuploaded firmware using link provided in MakerFarm guide (BTW, should I try vanilla Marlin firmware, will it make any difference?), enabled EEPROM access just in case. Then I decided to try printing 20mm 100% infill cube, as per gmay3's post. I used adjusted E-steps, which I saved to EEPROP and I also set Extruder Multiplier in Slic3r to 1.1 just to see what will be the result.

Here's the result:
IMG_8013.jpgIMG_8015.jpg

I've tried two times, and both times the print started bending around corners, and then it lost adhesion (the first time this happened for me). At this point I gave up.

Looks like something is definitely off with extruder, since even Extruder Multiplier doesn't force it to extrude more material. I'm not really sure what to try now.