I've always thought the best way to learn is by asking stupid questions to smarter people

If by smarter you mean more knowledgeable in any given subject than it is a really good way how to learn stuff, but I probably wouldn’t go as far to say that it is the best one :P.

The problem that I have with mathematical and physicalist definitions of information is that they shift an enormous amount of the explanatory weight onto the question of relevance…

Do you mean that they ignore the relevance and context? In that case yes. It's story not for another field, but different paradigm or epistemology for different day. Do not try to rush things young padawan. One step at a time! (will be further explained below)

Unless the relevant information is somehow sifted from the irrelevant information (relevant from the perspective of the receiver/interpreter) then I don't see how the mathematical model really gets us anywhere all that deep and interesting.

The information on its own is not irrelevant for some fields. It has helped and increased the speed in which internet grew all around the world in the (more, or less) efficient way it works up to date. It has also helped the cryptography and by that helped to unable the birth of cryptocurrencies. Now that is fucking relevant and deep to me even though it was all made possible (or just its origination was accelerated) due to (using your words) model that lacks the relevant information, therefore won’t get us anywhere deep. Just because the information lacks the contextual meaning doesn’t mean that the technical data cannot be used effectively. You’re correct that hashing and decoding and other cryptography stuff needed something “else” to work, but without the first steps there wouldn’t be the next ones, or they would lack the background and such a model would need to be “researched” retrospectively.

I see that you didn’t have a problem with the biological model. Is it because you sniffed Darwin in there? :P If that was the case you were correct. :D