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Yes. All of your examples trigger when you cast a spell. They happen whether that spell resolves or not.
I have seen multiple EDH decks built specifically around cards such as those and Cipher, where the intent is to exile spells with Nivmagus whenever you don't really need them.
Yes, because you cannot exile the spell with Nivmagus Elemental until the triggered abilities have already been placed on the stack (and if you want, resolved).
From the comprehensive rules:
As soon as you cast the spell (pay costs, select targets, place the spell on the stack) the triggered abilities are placed on the stack.
Steps 601.2a-g have been omitted and involve selecting legal targets and paying costs for the spell.
After the spell and triggered abilities are on the stack & you get priority back (either by already having it, or having your opponent pass it to you). You can then activate Nivmagus Elemental's ability. Because the triggered abilities are already on the stack they will resolve even though the spell that triggered them does not resolve.
The question doesn't make any sense because the casting of the spell and therefore the triggering of Nivix Cyclops's ability happens before you even have a chance to activate Nivmagus Elemental's ability.
Nivmagus Elemental's ability requires that the spell already exist on the stack, which means it requires that the spell must already have been created through casting or copying before it can be activated, which means Nivix Cyclops's ability would already have been triggered and would already have been placed on the stack by the time you can use Nivmagus Elemental's ability. In fact, you can even let Nivix Cyclops's ability resolve before you activate Nivmagus Elemental's.
Nivmagus Elemental's ability won't rewrite history so Nivix Cyclops never triggered, and it won't remove Nivix Cyclops's ability from the stack. As such, it won't stop Nivix Cyclops's ability from resolving.