Does "neues" have the "es" ending because (a) it is strong inflection because there is no article for "Auto," (b), this is in the nominative case because of the verb "ist," and (c) "Auto" is neuter (das Auto)? That is my suspicion.
Yes on (b) and (c), but it’s not stong inflection – although it doesn’t make a difference in this particular case. Possessive pronouns, like the indefinite article, trigger mixed inflection (strong endings in nominative and accusative singular, otherwise -en).
Why is it -er ending on unser? I thought nominative neuter endings were -e... I was doing so well until this one!!
-e in nominative would be feminine: unsere neue Küche. Or plural: unsere neuen Autos.
In nominative neuter should it not have been "unseres" and "unserer" for masculine?
Not as a possessive determiner (before a noun).
For some reason, those inflect like the indefinite article ein and have no ending at all for masculine nominative, neuter nominative, and neuter accusative.
(As a possessive pronoun, replacing a noun, those endings would be correct. Dein Pferd is groß und unser(e)s ist klein; dein Hund ist schnell und unserer ist langsam. "Your horse is big and ours is small; your dog is fast and ours is slow." Even English has this distinction: no ending in possessive determiners, e.g. "our"; ending in possessive pronouns, e.g. "ours".)
Possibly. Or perhaps it was a "type what you hear" exercise?
Hard to say anything more without a screenshot.
Because attributive adjetives (roughly: ones before a noun) need an ending on them. You can't just put the base form neu before a noun like that.
Which ending depends on the gender, number, and case of the noun, as well as on whether you have a determiner in front of it and if so, what kind.
See e.g. https://en.wikipedia.org/wiki/German_adjectives for more information.
Here, you have neuter nominative singular, mixed inflection after possessive unser, and thus you need the ending -es on neues.
German der, die, das are basically "that" but are also often used for "this" -- the distinction there is not as strongly made.
A bit like how English doesn't really distinguish between "that" and "yon" these days but uses "that" for both of them.