I have read that the statement "There is no absolute truth" is self-refuting because it relies on absolute truth to be true. I have also read that the idea expressed in the previous statement commits the fallacy of begging the question. I am thoroughly confused by the debate here...?

Relativists do seem to be in trouble with having to live with a relative notion of truth for their own claims. But I am not sure that Allen's worries are decisive. Suppose the claim is that there are no absolute truths, but there are truths relative to standards that you and I accept. You can try insisting that you adhere to a valid standard according to which my claim is false. But then we could argue about whether that is right. Insisting is one thing, being rationally persuasive is another. Similarly, I might claim that a given proposition really will be true by some standards and really will be false by other standards and further say that that claim itself is true relative to standards that we both accept .. but false by others. This sort of relativism might not be motivated by a desire dogmatically to insist that one has a right to one's own opinion ... it might be motivated by deep philosophical views about the relationship between language and/or thought and reality ... I think this...

One can create axioms that make statements like "all bachelors are married" true. What is wrong with calling these truths analytic as a shorthand for the type of truth it is based on the type of axiom it is derived from, much in the way we use the adjectives arithmetic, set-theoretic, or logical to denote those types of formal truths? I feel like one could decide whether a truth is analytic by seeing which (kinds of) axioms need to involved in making it true.

I don't know what you mean by: "One can create axioms that make statements like 'all bachelors are married' true". I assume that by 'married' you mean 'unmarried'. But I still don't understand. Perhaps you mean that one can write down obviously true principles from which the truth of every analytic sentence, and no other sentence, follows. But we can't. Not yet anyway.