Context. Or even the lack thereof. In this case the use of the conditional implies an unstated condition, the "if I'm right about it" condition.

This is more idiomatic than grammatical. It's the same in Spanish. In English we don't have this sort of idiom, so that phrase doesn't translate very well.

How would that be phrased in Spanish? Are you talking about the future tense in constructions like "Será que...?" or "Estará volviendo" to say "he must be returning"?

Julia habría dormido con Pedro anoche.

As explained, mood is implied. This is a common pattern in news reports, etc., not quite colloquial

Yeah, it's very common in news because it's a very concise way to indicate the possibility of doubt because it's just "gossip" (as the title here would have it).

Yes we do, “…will have slept…”

Merci.

What you say is grammatically correct. But if you want to find out whether it implies gossip or past conditional, you need to look at the context.