The copies that replicate creates are created on the stack, so they're not "cast." Abilities that trigger when a player casts a spell won't trigger.

The replicate triggered ability that creates the copies can itself be countered by anything that can counter a triggered ability. If the ability countered, no copies will be put onto the stack.

A copy of a spell can be countered like any other spell, but it must be countered individually. Countering a spell with replicate won't affect the copies.

As the replicate triggered ability resolves, you'll copy Changing Loyalty for each time you paid its replicate cost, even if the original spell is no longer on the stack at that time (perhaps because it was countered).