Learn Croatian: Nemoj ubaciti novac u ovaj aparat. - Do not insert money into this machine.

You are right that negative commands usually take imperfective verbs (like nemoj ubacivati) to mean "don't do that activity in general." However, using the perfective ubaciti turns the command into a strong warning against making a single, specific mistake, like "be careful not to insert" or "don't make the mistake of inserting."

Why do "ovaj" and "aparat" look like they are in the nominative case after the preposition "u"?

They are actually in the accusative case! The preposition u takes the accusative when indicating motion or direction (into the machine). Because aparat is a masculine inanimate noun, its accusative form is identical to its nominative form, so the pronoun ovaj also stays the same.

If I wanted to warn a group of people, or someone I address formally, how would this sentence change?

You would only need to change the negative imperative word to its plural or formal form: nemojte. The rest of the sentence remains exactly the same: Nemojte ubaciti novac u ovaj aparat.

Is "novac" in the accusative case here, and does it have a plural form?

Yes, novac is the direct object so it is in the accusative case. Just like aparat, it is a masculine inanimate noun, so its accusative form matches the nominative. In Croatian, novac is treated as a singular mass noun (just like "money" in English), so you don't use a plural form here.