You can think of Solomonoff induction as a weighted set of experts. At each step, each expert makes a prediction, and the overall prediction is the one that gets the highest total weight. Then the true data comes in, and the weight of each expert is increased or decreased based on whether that expert was right or wrong. And here's the kicker, the experts are all possible prediction programs, initially weighted by 2^-(program length). So yeah, in the long run Solomonoff induction will be at least as good as you at predicting any particular sequence, including prime numbers etc., because your own prediction algorithm is somewhere in the Solomonoff mixture. That also explains why approximating Solomonoff induction takes a huge amount of computation time. It's mostly a theoretical idea.
The output of a program could be infinite and thus it never halts. Without Chatlin's constant or the Busy Beaver values, brute forcing is not feasible in a countably computable universe. It is still interesting to talk about Oracles, ie. Somehow getting hold of Chatlin's constant and thereby easily solving the halting problem and being able to use the induction.
These are easily fixed by dove tailing through the programs and using an increasing-over-time cutoff. Not fixed in the "made practical" sense, but in the "made non-contradictory" sense.
Solomonoff induction goes from program to output to prediction weighted by program's length. It never goes from type of prediction to shortest equivalent program. It doesn't need to compute Kolmogorov complexities.
(Well, more specifically, approximations to it with time cutoffs don't have to compute Kolmogorov complexities. Raw Solomonoff induction already trivially runs into the halting problem.)
