I know that the construction "It is not so much funny as interesting" is valid if I want to talk about something that is both funny and interesting, but with an element of comparison.
To me, that sentence rather suggests that something isn't funny. I might use it after I've called something funny, and someone disagreed, and now I'm hurriedly backpedaling.
Is it still correct if I add "it is", as in, "It is not so much funny as it is interesting"?
Yes, that's fine. I might suggest a slightly different word order in that case — "It's not funny so much as it is interesting" — but I wouldn't blink at your version.
Is it possible to rephrase this sentence using "less"? For example, "It is less funny than [it is?] interesting"?
Yes, that's fine (either with or without the "it is").
If the bullet point above is correct, then would I need to use "more funny" or "funnier" if I were to turn the sentence around? I know that the correct comparative is "funnier", but it sounds unnatural to me in this sentence. But this point is irrelevant if the entire construction is wrong.
"It's more funny than interesting", or "It's more funny than it is interesting", is absolutely fine. This construction is sometimes called a metalinguistic comparison or metalinguistic comparative, because it's a bit like saying "The word 'funny' applies to it better than the word 'interesting' does." It's not the same kind of comparison as "This is funnier than that is."
That said, I've sometimes encountered the one-word comparatives in such cases; I don't think it's anywhere near as common, and I share your feeling that it sounds unnatural, but some native speakers do produce it.