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Inspired abilities trigger no matter how the creature becomes untapped: by the turn-based action at the beginning of the untap step or by a spell or ability.
Although the permanent you control can't be a land, the other target can, provided it shares a card type with the first target. For example, you could target an artifact you control and an artifact land controlled by an opponent.
The relevant card types (that is, the ones a nonland permanent can have) are artifact, battle, creature, enchantment, kindred, and planeswalker.
If another spell or ability allows you to change the targets of the ability (or perhaps copy it and choose new targets for the copy), you can change the targets only such that the final set of targets is still legal. For example, if the ability targets a creature you control and a creature an opponent controls, you couldn't change just the second target to a noncreature permanent controlled by that player. You could, however, change just the second target to a creature controlled by a different opponent. The two new targets can share a different card type than the two original targets did.
If the inspired ability includes an optional cost, you decide whether to pay that cost as the ability resolves. You can do this even if the creature leaves the battlefield in response to the ability.
If an inspired ability triggers during your untap step, the ability will be put on the stack at the beginning of your upkeep. If the ability creates one or more token creatures, those creatures won't be able to attack that turn (unless they gain haste).
Gaining control of an Aura or Equipment doesn't cause it to move, though gaining control of an Equipment will allow you to activate its equip ability to attach it to a creature you control later.
The exchange of control lasts indefinitely. It doesn't expire when Daring Thief leaves the battlefield.
If the full exchange can't happen, perhaps because one of the targets is illegal as the inspired ability tries to resolve, then nothing happens. No permanents change controllers.
Inspired abilities don't trigger when the creature enters the battlefield.
