When can I use "any..." in affirmative sentences?

Solution 1:

Any and its compounds are Negative Polarity Items (questions are Negative Triggers), though that's probably not what a grammar table would tell you. But that's not the only use of any.

There is also a different sense of any, with different grammar. It's called "Free-Choice any" and it belongs to that set of quantifiers like each, every, and all. Whereas all means 'complete group, taken together', and every means 'complete group, taken individually', and each means 'an individual, as part of a group', free-choice any means 'choose an individual from a group at random'; sort of a syntactic Axiom of Choice.

This any is normally restricted to use with modals of possibility,
i.e, it's a Possible Polarity Item like tell time:

• Anybody could do that is grammatical, but not *Anybody did that.
• Jimmy can already tell time is grammatical, but not *Jimmy already told time.