If Conduit of Ruin or any other card states that when it is cast, does this mean that it can trigger when it returns from the battlefield? Such as Conduit of ruin is in the graveyard, I return it to the battlefield from the graveyard. Does the first ability trigger and allow me to search my library for an eldrazi card 7 or more to place on top of my library?
No, "casting" a spell has a more specific meaning
What does casting mean?
601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. [..]
Normally, you are only allowed to cast spells from your hand, unless a card says otherwise. There are cards that allow casting spells from somewhere other than the hand, such as Bring to Light (casting from exile), Descendants' Path (casting from library), or Karador, Ghost Chieftain (casting from graveyard). The Commander game variant allows you to cast your commander from the Command zone.
Casting a spell or ability with a reanimation effect such as the namesake Reanimate never counts as casting the reanimated creature itself. It is simply an effect of that spell or ability.
If you are returning the Conduit from the graveyard straight to the battelfield as a result of another spell or ability (such as Reanimate), then it will not trigger, as 'when you cast' triggers are not the same as 'when this enters the battlefield' triggers.
If, on the other hand, you played a card that specifically allows you to cast a card from your graveyard (such as Havengul Lich), then it would trigger, as you are actually casting it.
No. "When it is cast" means "when the cast keyword action is performed to it". To cast something involves putting it on the stack, paying costs and making some choices. It has absolutely nothing to do with the battlefield, and even less with returning (putting) on the battlefield (which is an unrelated keyword action).