"These are not his cookies!"
Translation:To nie są jego ciasteczka!
"To jest/są X" and it's negative "To nie jest/są X" are constructions that take the Nominative for X.
Here we just have the nominative plural for cookies = ciasteczka
the jego bit is easy, jego doesn't have to match gender case or number when it means "his" (it has other meanings too, but here it means just "his").
Everything you say is right but you meant "jego",
"tego" means this/the and unfortunately has to match case gender and number.
Here, "these" is not a determiner, but a dummy pronoun, serving as the subject of the sentence. In Polish it will always be "To", used for "This/That/It is" and "These/Those are".
"te" would work in "These cookies are tasty" = "Te ciasteczka są smaczne", where "these" is a determiner.
'him' can be translated, depending on the place in the sentence, either as 'go' or 'jego'.
'his' can be only translated as 'jego'. I guess it's better to treat it as two separate words that are just accidentally identical.
Is this an exception to when the genitive is required? I would say this is the negative version of 'They are his cookies' so would have thought genitive would be the correct case to use.
Genitive is required for negation, only when the positive sentence had Accusative. Accusative is probably the most widely used case, so learners just get used to 'every negation = Genitive', but that is not true. The other cases do not change when negated. Negated Instrumental is still Instrumental, negated Genitive is... well, Genitive, etc.
The sentence here is basically just a simple "This is X" (or rather "These are X") sentence, so it uses Nominative and stays in Nominative when negated.
Well, let's start with the fact that this is not the Accusative pronoun. This is simply the possessive "his" and it is always "jego", in every case (yay, something doesn't change in Polish! also "jej" = "her" and "ich" = "their" don't change).
But answering the question anyway, "go" is the basic, neutral form, while "jego" is emphatic. So for example "Lubię go" = "I like him", but "Lubię jego, a nie ciebie!" = "I like him, not you!".