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.
Use thrown finesse weapons and the Sharpshooter feat
Thrown weapons only require one hand to draw and throw. You can use your free object interaction each round to draw a new one each round. This just requires you to be carrying enough to get through a combat encounter or two.
As a rogue, most of your damage will be coming from Sneak Attack, so the smaller damage die of the weapon won't matter so much. You do need to use a finesse thrown weapon, however, to activate Sneak Attack. The best option for this is the dagger or dart. The dagger has the added benefit of being useful in melee, but the dart is cheaper and lighter, which makes it easier to pick up a bunch of them.
The dart and dagger both have range of 20/60, so if you are throwing them beyond 20 feet, you will have disadvantage and lose the ability to use Sneak Attack. The solution to this is the Sharpshooter feat, which allows you so throw up to the weapon's maximum range without disadvantage.
Alternative: Grab the Dual Wielder feat
The Dual Wielder feat will give you a +1 to AC while you are dual wielding, and allow you to draw 2 weapons per turn. Have a dagger in each hand and you can attack twice per turn (once with a bonus action) for only the loss of 1 AC compared to using a shield. It does cost a feat, but this can be a good option.
Everybody needs a martial weapon proficiency
The shield bash rules say this (emphasis mine):
Shield proficiency says this:
Shield proficiency doesn't give you any kind of weapon proficiency with shields. It gives you the ability to use the shield and not take the armor check penalty. In order to not take a penalty for using a shield bash, you need that martial weapon proficiency. Classes that give you all martial weapons take care of that pretty easily, like a level of Fighter (or Radiant Servant of Pelor if you're a Cleric).
What about "Simple Weapons Only"?
The other classes you mentioned actually have a list of what they're proficient in, the Cleric's "simple weapons only" is just shorthand for a list that has every single simple weapon on it. It doesn't mean that Shield Proficiency works differently for Clerics than it does for everybody else.
It's also worth noting that the Cleric says "all simple weapons", not "only simple weapons". It's not a special rule or exclusion, it's just a shorter way of saying it than listing every simple weapon.
Pathfinder - The Same
From looking around, this appears to be the same in Pathfinder. The shield bash rules are basically the same, the shield proficiency rules are worded a bit differently but don't affect this, and James Jacobs (Creative Director at Paizo) agrees.
In fact, the Paizo FAQ states clearly that shield proficiency != weapon proficiency with shields.