Do Countered Spells Count as Cast?

Yes, the spell is cast. This will count for "on cast" triggers, Storm, Aetherflux Reservoir, Approach of the Second Sun and Commander Tax.

Do cast triggers happen if countered?

Once on the stack, the triggered ability will resolve regardless of whether the original spell actually resolves. So you will get the Eldrazi "cast" effects even if the Eldrazi are countered.

What happens when a spell is countered MTG?

A spell that is countered is put into the graveyard instead of doing its effect. It is essentially negated. Counterspells or permission spells may or may not have conditions, such as forcing a player to pay an additional amount of mana.