Learn French: Cette ancienne civilisation a disparu il y a un millénaire. - This ancient civilization disappeared a millennium ago.

Yes, it does! But when il y a is followed by a period of time, it means "ago". So il y a un millénaire translates to "a millennium ago". Notice that in French, il y a goes before the amount of time, whereas "ago" goes after the time in English.

Why use cette instead of ce?

Civilisation is a feminine noun (a helpful tip: almost all French nouns ending in -tion are feminine!). Because it's feminine, we must use the feminine word for "this": cette.

Why is ancienne before the noun? Doesn't that usually mean "former"?

You're right that ancien before a noun typically means "former" (like mon ancien patron for "my former boss"). However, for grand historical concepts like cette ancienne civilisation, placing it before the noun to mean "ancient" is very common and gives the sentence a slightly more formal, storytelling feel.

Is disparu a regular past participle?

No, it's irregular. It comes from the verb disparaître (to disappear). In the passé composé (the past tense), it changes to disparu. It uses avoir as its helping verb, giving us a disparu (disappeared).