HexBlade Warlocks give you proficiency with shields:

Hex Warrior

At 1st level, you acquire the training necessary to effectively arm yourself for battle. You gain proficiency with medium armor, shields, and martial weapons.

Can a Warlock (with Warcaster feat) use both staff (only for casting, never melee attacking) and a shield to increase AC? and how to deal with staffs that gives you extra AC like Staff of Power

Staff of Power

DMG Staff, Major, Very Rare (Requires Attunement by a Sorcerer, Warlock, or Wizard) 4 lbs.

This staff can be wielded as a magic quarterstaff that grants a +2 bonus to attack and damage rolls made with it. While holding it, you gain a +2 bonus to Armor Class, saving throws, and spell attack rolls.

Can a warlock with

Shield +1 [+3 AC]

Staff of Power [+2 AC]

Half Plate Armor [AC 15 + Dex (max 2)]

Reach 15 + 2 + 2 + 3 == 22 flat AC? and have all the benefits given by the staff? (and he can still cast Shield spell to reach 27 AC)