Learn Swedish: Allting är huller om buller, och vi hittar ingenting. - Everything is topsy-turvy, and we find nothing.

Both allting and allt mean "everything" and function as standalone pronouns, so you could absolutely say Allt är huller om buller. The difference is that allting is exclusively a pronoun (it stands alone), whereas allt is a dual-purpose word that can also act as an adjective before an ett-word (like in allt smör, "all the butter"). You can never put a noun right after allting.

Does the exact same rule apply to ingenting versus inget?

Yes! Because ingenting stands alone here as the object of the verb ("find nothing"), it works perfectly. You could swap it for inget (...och vi hittar inget) without changing the meaning. But if you wanted to say "we find no food", you must use the dual-purpose form as an adjective (vi hittar ingen mat or inget bröd). ingenting can never attach to a noun.

huller om buller is a fun phrase. How exactly is it used in Swedish?

It is a very common fixed idiom meaning "topsy-turvy", "pell-mell", or simply "in a complete mess". It is mostly used to describe physical chaos, like a messy room where toys or clothes are scattered everywhere, but it can also describe disorganized schedules or situations.

Could I negate the verb instead and say ...och vi hittar inte någonting?