If I cast an eldrazi with Hide Away cards like Windbrisk Heights, I presume the on-cast trigger wouldn't work because I didn't actually cast it? (Similar to "play" copies of spells?)
You actually do cast the spell. It uses "play" instead of cast because hide away lands also let you play land. When play is used in this context it means to either cast a spell or play a land.
If I cast an eldrazi with Hide Away cards like Windbrisk Heights, I presume the on-cast trigger wouldn't work because I didn't actually cast it? (Similar to "play" copies of spells?)
Windbrisk Heights says "You may play the exiled card..."; abilities that trigger "when you cast" that card (such as those of Phantasmagorian or Ulamog, the Infinite Gyre) will trigger when the card in question is cast as a spell (C.R. 701.16b, 701.4a, 108.1). Note that the hideaway ability itself doesn't let you play any cards (C.R. 702.75a). Note also that the Eldrazi creature type has no abilities or game mechanics inherent to that type, including abilities that trigger when a card with that type is cast.
EDIT (Nov. 23, 2019): One rule was renumbered in the meantime.
EDIT (Jan. 24, 2020): Add rule citation.
EDIT (Jul. 26, 2020): One rule was renumbered with Core Set 2021.
EDIT (Apr. 20, 2022): Some rules were renumbered in the meantime.
Shu Yun, the Silent Tempest WUR Voltron Control
Temmet, Vizier of Naktamun WU Unblockable Mirror Trickery
Ra's al Ghul (Sidar Kondo) and Face-Down Ninjas
Brudiclad, Token Engineer
Vaevictis (VV2) the Dire Lantern
Rona, Disciple of Gix
Tiana the Auror
Hallar
Ulrich the Politician
Zur the Rebel
Scorpion, Locust, Scarab, Egyptian Gods
O-Kagachi, Mathas, Mairsil
"Non-Tribal" Tribal Generals, Eggs
http://www.mtgsalvation.com/forums/magic-fundamentals/magic-rulings/magic-rulings-archives/300038-emrakul-and-the-hideaway-lands
EDIT (Nov. 23, 2019): One rule was renumbered in the meantime.
EDIT (Jan. 24, 2020): Add rule citation.
EDIT (Jul. 26, 2020): One rule was renumbered with Core Set 2021.
EDIT (Apr. 20, 2022): Some rules were renumbered in the meantime.
Shu Yun, the Silent Tempest WUR Voltron Control
Temmet, Vizier of Naktamun WU Unblockable Mirror Trickery
Ra's al Ghul (Sidar Kondo) and Face-Down Ninjas
Brudiclad, Token Engineer
Vaevictis (VV2) the Dire Lantern
Rona, Disciple of Gix
Tiana the Auror
Hallar
Ulrich the Politician
Zur the Rebel
Scorpion, Locust, Scarab, Egyptian Gods
O-Kagachi, Mathas, Mairsil
"Non-Tribal" Tribal Generals, Eggs