What are the various ways to express nothing in Russian, and the rules that govern usage?
For example, I came across this sentence: Жена умерла, и ему больше нечем жить.
Why нечем here and not ничего?
Also, when is it appropriate to use ничто over its more commonly used genitive case, ничего?
This is a colloquialism close to "nothing to live for", yet its literal meaning is "nothing to live by" which is an Instrumental case, so here we use "нечем" which is an Instrumental case of a pronoun "нечего". On the other hand, ничего is Genitive case of ничто, so it shouldn't be used here by two causes: wrong pronoun and wrong case.
Considering the difference between ничто и нечего: "ничто" is like "void-object", while "нечего" is "no object". So "жить ничем" means "to live for Nothing, to live for Zero", while "жить нечем" is "to have no reason to live" (or, sometimes, "to have no money to live").
Concerning case usage "ничто" vs. "ничего", you may want to look at this question: Why “ничего” and not “ничто”?
Ничего (ничему, ничего, ничем, ни...чём) is used to denote the absence of something, it refers to an unidentified thing which doesn't exist or isn't present and is the accusative of ничто. It declines similarily to the corresponding interrogative/relative pronoun. When it is used with a preposition, the prefix ни- is detached and the preposition comes between ни and the second part of the pronoun (ни...чему, ни...чего, ни...чем). Note that ничто in the accusative case takes the forms of the genitive. When it is used in a sentence, the verb is negated with the negative particle не. Unlike English, Russian sententces can have several negatives together with a negated verb (ничего+ не+ глагол в личной форме).
Никто ничего не знает.(Nobody knows anything).
Нечего( нечему, нечего, нечем, не о чем) is used to denote the absence of an object for an action and is followed by an infinitive.
Нечего читать (nothing to read), нечего надеть(nothing to wear). The English structure will be
There's nothing to do. Or They have nothing to do.
Besides нечего is used in some idioms
(нечего говорить--it goes without saying)
Your example belongs to idioms too.
Ему больше нечем жить.(He has nothing to live for).
Нечего can also be used in orders with negative meaning.
Нечего тут сидеть! - There is no good in sitting here.
However, this negative pronoun does not occur in the nominative case form to express negation, since impersonal consructions in which it is used do not have subject. The nominative form may occur ( нeчто), but it will have a different meaning and refer to an unspecified objects(see Indefinite Pronouns).
When it is used with a preposition, the prefix не- is detached and the preposition is placed between не and the second part of the pronoun.
(Мне с вами не о чем говорить.)
Examples.
Почему вы ничего не читаете?--Потому что мне нечего читать.
This must have roots in the Church-Slavonic:
"Не хлебом единым жив человек." - A man does not live by bread alone.
Notice that "хлебом" is in the Instrumental case.
In modern Russian you can ask "Чем он живет?" which can be translated as "What does he live for?" or "What is important in his life?"
