If a nonland permanent changes control while it's the target of Enigma Thief's ability, that permanent won't be a legal target as the ability resolves. Any remaining legal targets will be returned to their owners' hand.

To determine the total cost of a spell, start with the mana cost or alternative cost (such as a prowl cost) you're paying, add any cost increases, then apply any cost reductions. The mana value of the spell is determined only by its mana cost, no matter what the total cost to cast the spell was.

You can cast a spell for its prowl cost any time in a turn after a creature you control of a matching type has dealt combat damage to a player. It doesn't matter if that player left the game, if that creature left the battlefield or left your control, or if that creature no longer has a matching type.

Prowl compares the creature types of the spell with the creature types of the creatures that dealt combat damage to players this turn. Normally this means that you can cast Enigma Thief for its prowl cost only if a Sphinx or Rogue dealt combat damage to a player, but if an effect causes the spell to have other creature types, the prowl ability is also satisfied by a creature with those additional types having dealt combat damage to a player.