Precursor Golem

The copies that Precursor Golem's second ability creates are created on the stack, so they're not "cast." Abilities that trigger when a player casts a spell (like Precursor Golem's second ability itself) won't trigger.

Any Golem that couldn't be targeted by the original spell (due to shroud, protection abilities, targeting restrictions, or any other reason) is just ignored by Precursor Golem's second ability.

If the spell that's copied is modal (that is, it says "Choose one —" or the like), the copies will have the same mode. Their controller can't choose a different one.

The second ability triggers whenever a player casts an instant or sorcery spell that targets only one Golem and no other object or player. That Golem can be Precursor Golem itself, one of the Golem tokens it created, or any other Golem. It doesn't matter who controls the Golem.