How to assess the strength of the avalanche effect for AES?
Does it matter WHICH bit I'm flipping? Is it legit to always flip the MSB f.e. or should it be a random position?


If you see the cipher design as a black box then certainly yes. It could well be that the design would be more secure for certain positions than the other. This would indicate a (local) weakness.

Note that the Hamming distance is just one method of testing (pseudo) randomness.


  The presence of the Avalanche effect implies for an attacker to have a 50/50 chance to predict the input (I) state of one bit for each round. So the benefit of applying several rounds (in my case b/k=128 bit -> 10 rounds) is to reduce the probability for an attacker to predict a bit's original (plain) state to 0,5^10?


No. Each bit has two states. This means that one state is correct and one state is incorrect. If you can "predict" the state with a chance of 0.5 then you're perfectly secure.

Guessing a bit right with a chance lower than 0.5 actually doesn't make sense at all. It means you guess wrong with a high probability. In that case you just have to invert the bit to get a higher result (of $1 - P$ where $P$ is the probability).

The additional rounds are needed to protect against other attacks than simply performing black box testing. AES has been "broken" up to 7, 8 or 9 rounds for 128, 192 and 256 bits respectively.