No, Cloudshift is resolved as a single, atomic event. There is no time in between the parts of its effect in which anyone has a window of opportunity to cast a spell.
From the Comprehensive rules:
116.2e Resolving spells and abilities may instruct players to make choices or take actions, or may allow players to activate mana abilities. Even if a player is doing so, no player has priority while a spell or ability is resolving. See rule 608, "Resolving Spells and Abilities."
"To respond to X" means "to cast or activate after X is placed on the stack but before X resolves."
One can respond to Cloudshift, but not to "bringing creature back". That's not a spell, activated ability or triggered ability. You do not get priority in the middle of a spell resolving [CR 116.2e].
If you cast Day of Judgement in response to Cloudshift, it you would resolve before Cloudshift does anything at all. Cloudshift cannot save a creature from Day of Judgement, but Fiend Hunter and Sudden Disappearance could.
I would say no, Cloudshift removes and puts back a creature as part of the same effect, so you cannot 'inturrupt' the effect in the middle of its action by casting some other spell. You could Day of Judgement before you cast or after you cast cloudshift, but you cannot remove your own creature, Day of Judgment, and then put it back.