Hexblade warlocks gain proficiency with shields at 1st level from the Hex Warrior feature (p. 55):

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 the War Caster feat (PHB, p. 170) use both staff (only for casting, never melee attacking) and a shield to increase AC?

And how to deal with staffs that give you extra AC, like the Staff of Power?

Staff, very rare (requires attunement by a sorcerer, warlock, or wizard)

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? (The shield spell would also temporarily boost this to 27 AC.)

Would this all work as I have described?