Heinner OMX-600 blender motor not turning
I do understand the dangers of working with mains voltage
  and would like to further expand my knowledge in this area.


Please go to school for this.  Making a mistake with mains is deadly.
Part of your schooling would be training so that you can preserve your life.
Proceeding without schooling is like Russian Roulette --
it's just a matter of time until you make a mistake, 
and feel the lightning bolt.  It is wise to learn from the wise.
Seek, value, and retain wisdom!  It will preserve your life!
And don't forget to listen to wisdom, once you find it!


  I was thinking of first ruling out that the motor is not faulty. 
  Where should I probe with my DMM on it? 
  Should I be measuring continuity(resistance) along the coils


This is a good idea. It looks like your rotor may be burned, as it 
is darker, which is not a good sign, and indicates an area to 
look for further problems, like short-circuits, or open windings.
The resistance reading should all be similar for symmetrical windings.
A winding that looks like the others, but has a significantly lower 
or higher resistance is likely one part of the winding shorted to 
another, or open altogether.


  or try to measure AC with it ON as so to make 
  sure it's getting power? ( Yes, dangerous if not careful)


Please do not do this!  There are other ways to make sure
that it's getting power.  Just measure the resistance. 
It should be fairly low.  But keep the power off while working!  PLEASE!
Only turn the power on to test for success.
And make sure to discharge any dangerous capacitors each time you unplug, 
and before you work on it. (Yes, it's better to do it twice and be safe.)


  Is there an practical way to test that the triac 
  or diode ( based on the datasheet I see it's not a simple zener diode ) is ok?


This I don't know.  But the triac is cheap, so order a few and 
swap them in.  For less than 2 usd, here's one:  "BTB06-600CRG -  TRIAC, 6A, 600V, TO-220AB"  Order at least 4.  If one winding is shorted, 
it might work for a little bit and then burn out the triac, 
so you may literally burn through a bunch of them.

You may also look at the brushes (spring loaded pieces of graphite) 
and make sure that they are contacting properly, 
and conducting electricity. 

EDIT:

Be careful when you're testing whether the motor works, to secure the motor, as it will "jump" and could damage or injure itself, you, or something else when it does.