The multiple meanings of "any" in different circumstances

Solution 1:

The first thing to know about any is that it's a Negative Polarity Item. That means it's got weird grammar and semantics, and is part of a lot of idioms, so one can't make assumptions about what it means. Negation is extremely complex.

The second thing to know about any is that it's a Quantifier, and works like other quantifiers in some ways. It doesn't do Q-Float, like each or all do:

• Each/All of the boys passed = The boys each/all passed.
• Did all of the boys pass? = Did the boys all pass?
• Did any of the boys pass?, but not *Did the boys any pass?

And it alternates with some, and -- to the extent it has any meaning, which is not very much -- any has the same meaning as some, which is also a quantifier, but is not a negative polarity item and therefore can occur anywhere:

• I saw some people coming this way, but not *I saw any people coming this way.
• He insisted I drink some wine, but not *He insisted I drink any wine.

even when any would be allowed, as in a question:

• Have you had something to eat yet? ~ Have you had anything to eat yet?

There is no difference between these two questions, although some speakers and addressees may have their own conventions for using them -- and any associated intonations.

As for the examples in the question:

• I have more chocolate than any of you.

is a use of any in the negative context of the baseline for a comparative construction (identified by than, which occurs only in comparatives, and therefore marks negation).

The second example is ungrammatical, as noted. Corrected, it would be

• Are there any of you with more chocolate than me?

which is a use of any in the negative environment of a question, like the some/any example above.

As soon as you notice that negatives keep popping up in example sentences, your first thought should be that there's an NPI involved.