Two questions, one rule each.

kein negates a noun that would otherwise take an indefinite article (ein/eine) or no article at all — uncountable nouns like Wasser, mass nouns like Geld, and zero-article plurals. nicht negates everything else: verbs, adjectives, adverbs, and nouns that carry a definite article (der/die/das) or a possessive (mein/dein/sein). The deciding test is simple: would the noun take ein/eine or no article? → kein. Anything else? → nicht.

The single unifying rule: nicht goes as late as possible, except directly before the element being focally negated. If you are negating a specific complement — a predicative adjective (nicht rot), a prepositional phrase (nicht nach Berlin), a separable prefix (nicht an-), or a participle/infinitive in the verb bracket (nicht gesagt, nicht kommen) — nicht lands directly before that element. If nothing specific is being focally negated, nicht falls at the end of the clause or just before the verb-bracket element. Most learners memorise seven separate position rules; they are all consequences of this one principle.

kein follows the ein-declension exactly: keinen Mann (Akk. masc.), keine Frau (Nom./Akk. fem.), kein Kind (Nom./Akk. neut.), keiner Frau (Dat. fem.), keinem Mann/Kind (Dat. masc./neut.), keines Mannes/Kindes (Gen. masc./neut.). The crucial difference from ein: kein has a plural form (keine Kinder, keinen Kindern in Dative), whereas ein has no plural at all — that is why German needs two separate negation words in this slot.

nichts is a pronoun meaning "nothing": Ich weiß nichts. nicht is an adverbial particle meaning "not" that negates something else: Ich weiß es nicht. They are not interchangeable. Additionally, nichts can take adjectival endings after it: nichts Neues (neuter nominative weak ending), nichts Wichtiges. nicht never takes noun or adjective endings.

Yes — and this is one of the most frequently missed points in beginner grammar. Ich habe keine Bücher means "I don't have any books." The plural nominative/accusative form is keine; the plural dative form is keinen (Ich helfe keinen Kindern). English speakers often over-use nicht here because they translate from "I don't have books" and reach for the particle instead of the negative article.

Total negation means the entire clause is negated: Ich gehe nicht ins Kino — "I'm not going to the cinema." Partial negation means only one specific element is negated, often followed by a sondern-clause that states the alternative: Ich gehe nicht ins Kino, sondern ins Theater — "I'm not going to the cinema, but rather to the theater." The position of nicht shifts accordingly: total negation → nicht falls late (often clause-final or before the verb bracket); partial negation → nicht sits directly before the focally-negated element. This scope distinction explains the entire position system.

nicht goes between the conjugated verb and the separable prefix at the clause end: Ich rufe sie heute nicht an — "I'm not calling her today." The prefix an counts as the second part of the verb bracket (Satzklammer), so nicht slots in just before it. Placing nicht elsewhere (*Ich rufe nicht sie heute an, *Ich rufe sie nicht heute an with the meaning of negating heute) produces a different scope — negating heute rather than the verb phrase — which is usually not what is intended.

Yes: nie / niemals ("never"), niemand ("nobody" — declines like kein in singular: niemanden, niemandem, niemandes), nirgends / nirgendwo ("nowhere"), weder … noch ("neither … nor"), nichts ("nothing" — a pronoun, covered above), and kaum ("hardly" — semantically negative but not a formal negator). Crucially, German does not use double negatives: Ich habe nichts is correct; *Ich habe nichts nicht is ungrammatical. Choose one negating element per clause.