The Circle of Loyalty's triggered ability won't trigger when you cast it because it's not on the battlefield yet.

The cost reduction ability reduces only the generic mana in the relic's cost. The colored mana must still be paid.

The Circle of Loyalty's triggered ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying, add any cost increases, then apply any cost reductions. The mana value of the spell remains unchanged, no matter what the total cost to cast it was.

Once you announce that you're casting a spell, no player may take actions until the spell has been paid for. Notably, opponents can't try to change by how much a relic's cost is reduced.