The note in the lesson says:

"This rule does NOT apply to sentences using the verb être and other Verbes d'état, with which the indefinite article doesn't change."

Note that the rule clearly specifies verbes d'état, of which rester is one. So the exception also applies to rester.

Animal familer means a familiar animal or a domestic animal while animal de compagnie means rural animal or bush animal

The voice sound sample for "animal de compangnie" still says "animal familier" just fyi.

The question "how come" is always tempting to ask and often difficult to answer, simply because languages aren't math and don't always follow the rules of logic.

In this case, it may help that English, too, doesn't use the definite article after negations. If someone asks you, "Do you have a sister?" you answer with, "No, I don't have a sister," and not "No I don't have one sister". In French it is similar.

Je ne fais pas du ski

du

de la

Yes, after être or other verbes d'état (there's a link to the full list in the lesson).

For example:

"Ce ne sont pas des choses aisées à dire" - "these are not easy things to say"

It's also used in some figurative expressions:

"ça ne casse pas des briques"

"les chiens ne font pas des chats"

no