Frightener artifact doesnt have doomed but for whatever reason cannot be summoned from the graveyard. Phoenix hatchling, a similar artifact has doomed and the doomed symbol to indicate the artifact is removed once used but Frightener does not so Caretaker should be able to summon the artifact
The artifact transforms into the immune unit. There is no artifact anymore for Caretaker to play.
Yes, however the artifact is still a card, Phoenix hatchling also transforms from artifact to monster but the card has a doomed symbol and says it is removed once changed. If Frightener works in the same way it should be stated on the card or just add the doomed logo to the artifact so we know it's removed from the game.
No, if artifact transforms into a unit, the artifact is gone. Doomed tag on Phoenix Hatchling means only, that if the artifact would reach the graveyard (by being destroyed, or at the end of the round), it would get banished.
The same way, if you transform a unit into a Kaedweni Revenant with Draug, you can't later resurrect the unit that was transformed into the Revenant.
The same way, if you transform a unit into a Kaedweni Revenant with Draug, you can't later resurrect the unit that was transformed into the Revenant.