QIf I discard this card during my end phase due to my hand being over the card limit, do I still draw a card with its effect?

AYes. However, you would still need to discard cards until you have 7 in your hand.

QIf I have 8 or more cards in my hand during my end phase and I discard this spell, can I draw a card with its ability?

AYes, you can. However, you would still need to discard cards until you have 7 in your hand.

QWhen I play this spell and it gets put into the cemetery, does its "When this card is discarded, draw a card" ability trigger?

ANo. Playing a card does not count as discarding it.