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?

You can use a staff and shield, because you can use staves one-handed anyway

A quarterstaff is a simple melee weapon with the versatile property, meaning that you can wield it one-handed without penalty - it just does slightly more damage if you use it two-handed:

Versatile. This weapon can be used with one or two hands. A damage value in parentheses appears with the property--the damage when the weapon is used with two hands to make a melee attack.

Since staves don't need both hands to use or wield effectively there's no problem with using both a staff and a shield. This also means that a staff is a valid weapon for the second benefit of the Hex Warrior ability (it lacks the two-handed property because you don't have to use two hands to wield it), so you could choose to use your charisma bonus to attack and damage rolls with it:

Whenever you finish a long rest, you can touch one weapon that you are proficient with and that lacks the two-handed property. When you attack with that weapon, you can use your Charisma modifier, instead of Strength or Dexterity, for the attack and damage rolls.

Even if you did need two hands to wield a staff properly, the Staff of Power states that you gain the benefit to AC, saves, and spell attack rolls by merely holding the staff, which is obviously possible to do with one hand. Similarly, you have the ability to cast spells from the staff while simply holding it, so you can hold it in one hand and cast spells from the staff using its charges without issue.

The items you've listed would all stack together in the manner you've described and give you an AC of 22, which you can indeed boost further by casting shield in an emergency.

Spellcasting with material components might still be tricky

With the War Caster feat, wielding a shield or weapon doesn't impede your ability to provide somatic components - but you don't have the ability to manipulate any material components for your spells, since you do strictly still need a free hand for that, and a hand holding a weapon or shield isn't free, even if you can still use it to make magical gestures. However, you don't have to provide most material components if you're using a spellcasting focus. A magical staff such as the staff of power might or might not count as an arcane focus.

If your DM rules that such staves are arcane foci, you can use the staff as a focus as it's held in your hand and you're good to go. If your DM doesn't count the staff as a focus, or you're casting a spell with a costly or consumable material component, you'd need to drop or stow the staff in order to manipulate your material components and then equip it again. (It's not feasible to stash your shield instead, because donning or doffing a shield requires the use of your Action.)

Luckily you can do all that in a single turn because dropping a held item requires no effort and you can then use your free object interaction on your turn to pick it up again after you cast the spell, but it does mean that you wouldn't get the staff of power's bonus to spell attack rolls when casting such spells, since you can't be holding it at the moment you cast the spell.

If you are a Pact of the Blade warlock (with the Hexblade patron that seems an obvious choice) and you take the Improved Pact Weapon eldritch invocation, you could turn a staff into your pact weapon and then use it as a spellcasting focus via the eldritch invocation, which lets you get the staff's benefit on most spells even if your DM ruled the staff was not inherently an arcane focus.

Yes

the Staff of Power does not say you need two hands to use it, so you can also wield a shield.

Even if you attack with the staff, it can be done with one hand.