No to all of these. The grand claims of that article are overblown hype (as is so often the case with New Scientist), and credit Chaitin with too much, to the exclusion of other mathematicians before him.

Anyone interested in Chaitin's work could read his own technical book "Algorithmic Information Theory", but might also read the criticism of him in Torkel Franzén's "Gödel's Theorem: An Incomplete Guide to its Use and Abuse" (book, not online, but reviewed here). The business in the original article of the hierarchy of Omegas is nothing more than the already well-known concept of degrees of unsolvability, which dates back to 1944.

Short answer to your questions: No. They aren't related.

Pedantic answer to your questions: Yes.

If you knew the value of Chaitin's Omega you could calculate the incalculable. You would know the results of computations that should take an infinite amount of time to calculate. You could summon the proofs of any conjecture. You could simulate AIXI. You would have the knowledge of a demigod. Being a demigod has strong bearing on the subjects you bring up, and many others.

Chaitin's Omega and LW's Omega share the same name (I suspect) because they both refer to something superhuman. They are different sort of superhumans, however.

Löb's theorem and Chaitin's constant are in the same family of weirdness, so I think it is likely relevant, but not for a 'new' reason, if that makes sense.

I can't comment on your question, but I have to say that that article is exceptional in managing to tell math research as an engaging story.