According to the Lopatin's reference:
§ 92. In the Genitive singular of the adjectives, participles and pronominal words of masculine and neuter gender, -ого (-его) is traditinally written with the letter г, but pronounced with в, for example: до́брого, хоро́шего, большо́го, си́него, четвёртого, чита́вшего, его́, чего́, того́, сего́, всего́, одного́, моего́. This also happens in the adverb ничего́ ('pretty good'), in the words ничего́шеньки, ничегòнеде́лание, сего́дня (and сего́дняшний), итого́; however, the word ничево́ки (name of the writing circle) is written with the letter в.
Note. In the word сеголе́тки (Nom. sg. сеголе́ток and сеголе́тка) г is not just written, but also pronounced.
Almost certainly because it's not close enough to "standard English" for Duolingo. Even if the extremely faint possibility of accepting the much maligned "ain't" were being considered (since I think it more frequently accompanies that unusual-to-English double negative), "jack" for nothing is probably too slangy to be worth including as a translation on Duo.
I'm not sure if it can be considered an exception. «Н-» gets appended to all pronouns beginning with е- if they are used after a pronoun and if they don't modify another word after them (so «у него», but «у его сестры»).
This is related to the history of the language, it doesn't really facilitate anything because «его» (/jɪ'vo/) starts with a consonant (/j/) anyway.
Well, historially it is similar to Celtic nasalisation, but it's limited only to a handful of pronouns.
Think of it this way: If Russian didn't put the Н in front of Его/Её, then how would you know which pronoun has which object?
Example: "У его игрушки кошки." = His toy has cats. (У takes the first visible genitive "игрушки.") "У него игрушки кошки." = He has the toys of a cat. (У takes the first visible genitive "него," which means the word after that must be a regular plural feminine noun. But wait, it's followed by another seemingly plural female noun, which must make THAT one another genitive singular feminine noun.)
The H serves a good purpose in these situations. :)
It's the genitive case in this sentence.
The nominative case is ничто, but it's a rare form that is usually replaced with genitive, at least in colloquial speech.
Yes. According to https://en.wiktionary.org/wiki/%D0%BD%D0%B5%D0%B3%D0%BE#Pronoun_3, " него" is genitive of both "он" and "оно́".