If paying the activation cost of the ability includes sacrificing Rings of Brighthearth, the ability won't be copied. At the time the ability is considered activated (after all costs are paid), Rings of Brighthearth is no longer on the battlefield.

The triggered ability of Rings of Brighthearth and the copy it creates both resolve before the ability that caused it to trigger. They resolve even if that ability is countered.

You can't pay more than once for each time the triggered ability of Rings of Brighthearth resolves.

An activated mana ability is one that produces mana as it resolves, not one that costs mana to activate.

If the ability has in its cost, the copy uses the same value of X.

The copy will have the same targets as the ability it's copying unless you choose new ones. You may change any number of the targets, including all of them or none of them. If, for one of the targets, you can't choose a new legal target, then it remains unchanged (even if the current target is illegal).