> I think half of the difficulty in any real world problem would be how to design the reward function.
That's exactly what MIRI's trying to do ;)
> do we really know that choosing the action that has the best weighted reward over the set on "world model algorithms" (hypothesis), produces actions that actually are intelligent?
As far as AIXI is concerned, this is the definition of an "intelligent action": that which leads to the largest expected utility over the agent's lifetime. It's fine to disagree with this definition, but one of the reasons to define AIXI at all is to have something concrete to point at, rather than spending decades debating these sorts of quasi-philosophical questions (note that I've carefully chosen words like "can be reasonably considered" rather than "is").
> I am thinking, maybe the shortest "world model algorithms", which are given the largest weight, are just mostly stupid.
The Solomonoff prior used by AIXI dominates all computable priors; in other words, even if these world models are stupid, no computable algorithm (including humans) can do better overall.
> And there is the No Free Lunch theorem, which states that averaging over all possibilities, while may sound clever, produces just garbage (i.e. no better than random guess).
The No Free Lunch theorem is a Mathematical curiosity with no particular relevance to the world. In particular, it completely ignores computational complexity: it gives equal weight to all (computationally) simple explanations (eg. "there is a star orbiting the Earth, it will keep orbiting" and "the Earth is spinning near to a star, and will keep spinning"), as well as to all (computationally) complicated explanations (eg. "the atmosphere has been bombarded by cosmic rays which, by sheer chance, have an effect which looks like a star, but it's unlikely for that coincidence to continue" and "the Earth is spinning near to a star, but tomorrow at 13:48 GMT the Martians, who have managed to elude all of our telescopes and probes, will attack the Earth with a weapon which switches the direction of rotation"), as well as all incomputable explanations. Trying to predict anything in such situations is clearly futile, which is basically what the NFL theorem says; yet such situations can never actually arise outside of though experiments.
Although AIXI itself is incomputable, it is specifically defined to interact with computable environments, so No Free Lunch doesn't apply.