Permanent spells that were countered earlier in the turn never entered the battlefield, so they won’t return to the battlefield because of the last ability.

Extra turns can still be created while Gerrard’s Hourglass Pendant is on the battlefield. They’re not skipped until they would begin, so if Gerrard’s Hourglass Pendant leaves the battlefield before that happens, the extra turns will be unaffected.

You choose what an Aura card put onto the battlefield this way will enchant. You can’t choose any permanent cards entering the battlefield at the same time as that Aura. If there’s nothing legal for the Aura to enchant, it stays in the graveyard.

An “extra turn” is any turn created by a spell or ability. Notably, it doesn’t include additional turns taken in tournaments after time expires for a round.

Permanent cards that were put into your graveyard from anywhere else, such as a card that was discarded, stay in the graveyard. Permanent cards that you gained control of and went to another player’s graveyard from the battlefield will likewise stay in the graveyard.