That's a lot of things packed into one.
You can also say я люблю свою маму, or ты любишь свою маму. Свой is the universal reflexive. It means "the sentence subject's own". Он любит его маму would mean "he (e.g. John) loves his (e.g. Pete's) mom".
он двадцать лет would be an unfinished statement: "for twenty years, he..." You can't say "he's twenty" this way for the same reason you can't say "him is twenty" in English.
Oh and by the way, ему один год.
Both его and него can be genitive as well as accusative. The н- has nothing to do with case; it appears after a preposition. The reason is similar to how "an eke name" turned into "a nickname", or "mine Ed" => "my Ned".
The Proto-Slavic prepositions *kъn "to" and *sъn "with" became *kъ and *sъ, but kept their old forms before third-person pronouns; this led to things like *kъn jemu "to him" and *sъn jimъ "with him" being reinterpreted as *kъ njemu and *sъ njimъ, and then, much like with "Ned" and "Nelly", it was generalised to every occurrence of those pronoun forms after a preposition, even the ones that didn't have that disappearing N.