If I cast a permanent again for 2 after airbending it, will its cast triggers be triggered again. Can I, for example, use [[Appa, Loyal Sky Bison]] to airbend [[Emrakul, the Promised End]], and take control of another player for the 2 recast cost? Or will even Emrakul’s cast cost reduction apply, allowing me to do this for even less?
Yes. There is a crazy infinite combo deck in standard now: [[Appa, steadfast]] [[aang, swift]] and [[doc aurlock]]. Here is a decklist for reference: https://mtgdecks.net/Standard/bant-airbending-decklist-by-harrison-fang-2782596