Can you Unmorph a manifest?

Yes. If a creature card with Morph is manifested, the permanent it becomes may be flipped for either its creature cost or its morph cost.

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Subsequently, is mtg manifest casting?

Does manifest count as casting a creature? No. Manifesting a card is always part of another spell or ability resolving. Then, can you manifest a land? So you can turn up a land, instant, artifact... Because the card is a manifested creature it is a creature card named manifest. If not then that means you can't identify your manifested cards that are not creatures(the first line).

Also, what happens when a manifest card dies?

When your manifested card dies, Gift's trigger tries to return that card to the battlefield. If it was an instant or sorcery: It want to return to the battlefield face-up, but that's simply unacceptable to the game rules for an instant or sorcery to be on the battlefield, so it stays in the graveyard, and so does Gift. Correspondingly, do manifest creatures have summoning sickness? No, it won't have "summoning sickness" unless it already had it. Turning the card face up ("morphing") just changes its characteristics, just like Giant Growth does. Turning the card face up doesn't give or take away "summoning sickness" any more than Giant Growth does.