expression
(not) anything; nothing (with negative verb)
Used with a negative verb to mean 'nothing' or 'not ... anything'. The negative verb is required.
冷蔵庫には何もない。
There is nothing in the fridge.
彼は何も言わなかった。
He didn't say anything.
expression
and everything else; and all (in ...も何も pattern)
Used in the pattern ...も何も to mean '... and everything else' or '... and all'. Emphasizes that the preceding item is included along with other related things.
仕事も何も全部放り出して旅に出た。
He threw away his job and everything else and went on a trip.
expression
(not) at all; (not) especially (with ...なくてもいい etc.)
Used with expressions like ...なくてもいい or ...ことはない to mean 'not at all', 'not especially', or 'not to that extent'. Softens the necessity or obligation.
謝ることは何もないよ。
There's no need to apologize at all.
急ぐことは何もない。
There's no need to rush especially.
何か means 'something' in positive sentences, whereas 何も is used in negative sentences to mean 'nothing' or 'not anything'.
Combination of 何 (nani, 'what') and も (mo, inclusive particle). The reading なにも is standard; なんも is a colloquial contraction.