Can I use "anymore" with "nothing"?

Solution 1:

Anymore (Any more) is a Negative Polarity Item (NPI), and nothing is a negative trigger word.

NPIs require a negative trigger word in their context, usually in a special syntactic construction.
It doesn't matter whether the clause is "positive", taken together, or not. It's the trigger that matters.

An NPI that occurs outside the scope of a negative trigger, however, is quite ungrammatical.
(though American English does have a "positive anymore" dialect, where anymore = nowadays).

• I doubt he goes there anymore, but not *I believe he goes there anymore.
• Few people go there anymore, but not *A few people go there anymore.
• Only Bill goes there anymore, but not *Even Bill goes there anymore.

(except, again, in positive anymore dialects, where people can say things like

• That's where I take my dry cleaning anymore.
  -- meaning "That's where I take it now; I used to take it elsewhere")