I wonder if anyone could help me understand exactly how the rumor 'Cursed Relics' works.

Is it so that it always rises the terror level by five, and then five more if you weren't able to sacrifice or exhaust any artifacts?

Every turn after it comes into play, the terror increase by one unless it is prevented. If not prevented, it will increase a second time at the end of the rumor.

So, it increase by 0, 2, 4, 6, 8 or 10, depending on how many Unique item you're discarding or Exhibit Items you're exhausting.

Right. The wording on the cards was incorrect in the first version of the expansion, but was not fixed in the revised edition.

When the rumor says "discard/exhaust an item to prevent both effect s ," it does not mean to prevent the doom token from leaving the rumor card: it means the token is returned to the box. So to pass the rumor you have to perform this action five times over five rounds, and the terror level will not have risen at all due to this card.