Generally speaking, yes, he could
Two-handed weapons require two hands, shields require one. If you have at least three, that means you can use both.
Shields do not have penalties for being in the “offhand”
That’s where they usually are, regardless of what weapon is otherwise wielded. Also, note that “offhand” doesn’t actually mean anything unless or until you choose to use Two-Weapon Fighting to gain an extra attack.
If you have more than two arms, you usually treat all hands except one as offhand. This again only matters for Multiweapon Fighting.
Tower Shields are really bad
Note that Tower Shields have lots of drawbacks: −2 to all attacks, huge Armor Check Penalty, tiny Max Dex, and they weigh a ton. They are rarely a good idea to use. I’d encourage your player to avoid it, not because it’s too powerful, but because it’s not powerful enough. A solid Heavy Shield gives half as much AC, but you ditch most of the problems you get with a Tower Shield. The Standard Action for Cover option is just not even remotely worth it.
And this is ignoring the fact that Aegides don’t get proficiency in Tower Shields, so he’ll have to multiclass or burn a feat to get it. That’s a big deal; feats should get you more than the option of using equipment that isn’t better than what you already could use.
The Aegis Customizations actually give two arms
As specified under the lesser extra arms customization, these options give a pair of arms, not just one. Note, however, that shields do not stack with one another, so that’s of limited benefit. Lesser extra arms can carry things, but not use them, so that is insufficient to actually benefit from any shield or weapon. Actually using a shield requires extra arms, minimum Aegis 5 and requiring a total of 3 points (2 points for extra arms and 1 point for lesser extra arms which is a prereq). That’s a pretty big chunk of a 5th-level Aegis’s available customization points.
Note that the fourth arm is difficult to use here. He could have a one-handed or light weapon in it, but unless he’s using Two-Weapon Fighting (and therefore has way more Dex than you’d want on someone who has a Max Dex of 2 on his shield, or is taking ludicrous penalties for TWF without the feat) he isn’t able to use that alongside his earthbreaker, so there wouldn’t be much point. Even then, it’s going to take another −2 penalty on attacks just for being in the extra arms; it takes greater extra arms (minimum 8th level, and 6 customization points) to avoid those penalties.
(If it comes up, TWF penalties are based on whether or not he has a Light weapon in the offhand; that his main weapon is two-handed does not affect these. Note he could do this without extra arms if he used Armor Spikes as his Light weapon. Thus, the probably-optimal way to use greater extra arms is Two-Weapon Fighting with a Two-Handed weapon, a Light weapon, and a shield, to minimize TWF penalties. But other than the shield, you could just use Armor Spikes to achieve the same thing, so this is not especially impressive.)
None of this is particularly overpowered
You have a 5th-level character with, as a sizeable portion of his accumulated class features, the ability to use a 2d6-damage weapon along with a shield. Note that a Rogue at this level could easily have 3d6 Sneak Attack damage, on top of a base weapon damage that could be 1d6 or more; sure, some things are immune to that (though quite a lot fewer in Pathfinder than in 3.5), and he has to do a little work to make sure it triggers (though Flanking is usually easy and to everyone’s benefit), but it’s still literally twice as much damage.
He does have more AC, plus the DR of the Juggernaut Armor, but those are his class features, and are a big part of the reason why he’s doing half as much damage, has half as many base skill points for a much smaller list, and so on. And if he actually goes for the Tower Shield, his huge penalties to attack and many skill checks are even bigger problems. Note that for the same penalty to attack, your typical Rogue could double his damage by using Two-Weapon Fighting, so now his damage is quadruple what the Aegis gets.
The Aegis gets to be really, really difficult to hurt with physical attacks. That’s his schtick, that’s what the class is good at. It’s a pretty narrow schtick and I worry that he’ll find that smart enemies just ignore him since he’s a pain to kill but has low damage output. Maybe he’ll invest other customizations towards improving that, but Aegides aren’t exactly amazing in that department; he’s never actually going to catch up with the Rogue who gets his dual-wielding full-sneak-attack. That’s the Rogue’s thing; this is the Aegis’s. I recommend against punishing players for being good at what they’re supposed to be good at.
Yes.
The actual rules for this are oddly worded, but the notation is pretty simple.
In general, there are two formats for "things that increase your AC." The most common format is the one attached to armor:
[your AC equals] 11 + Dex modifier
or Draconic Resilience:
your AC equals 13 + your Dexterity modifier.
Note that there is no "plus" at the beginning of the formula. These items set your AC to a specific value.
The other format is that used by shields and class features that increase your AC above its base value:
Defensive fighting style:
You gain a +1 bonus to AC.
Or a shield:
Wielding a shield increases your AC by 2.
These are phrased in terms of increasing, rather than setting your AC. They are preceded by the phrase "increases" or a plus sign.
So, it's pretty simple. You get one thing that sets your AC, and any number of things that modify it (except where restricted by other rules, like the rules stating you can only use one shield).
A fighter could very easily have an AC like this:
[15 + Dex Mod] (half-plate) [+2] (shield) [+1] (defensive fighting style)
For a total AC of 18 + Dex Mod.
But he couldn't add a chain shirt to increase his AC, because both that and the half-plate set his AC to a fixed value.
The same applies to Draconic Resilience. You can stack it with a shield, because a shield increases AC. And you can further stack on the Shield spell, giving you a (temporary) total of [13 + Dex Mod] [+2] [+5], or 20 + Dex Mod.
But you can't combine Draconic Resilience with Mage Armor, because both Mage Armor and Draconic Resilience are trying to set your AC to a fixed value.
Some things Draconic Resilience stacks with:
Some things it does not:
RAW
The rules for AC are on page 14 of the Player's Handbook, or page 9 of the Basic Rules.
The relevant part is this:
Without armor or a shield, your character’s AC equals 10 + his or her Dexterity modifier. If your character wears armor, carries a shield, or both, calculate your AC using the rules in chapter 5. Record your AC on your character sheet.
[...]
Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.
The rules called out above appear to be these:
Wielding a shield increases your Armor Class by 2. You can benefit from only one shield at a time.
If you wear light armor, you add your Dexterity modifier to the base number from your armor type to determine your Armor Class.
(and so on, for each type of armor)
The rules are sloppy here. Technically, they don't give you a calculation for AC when wielding a shield and no armor. And they never explicitly call out what is or is not a "way to calculate your AC."
With that said, we can use examples to build the intent of the designers.
It's pretty clear that you don't have to choose between a shield and plate. It's also pretty clear that a shield shouldn't set your AC to 2.
Finally, it's pretty clear that wearing a chain shirt with half plate shouldn't set your AC to 28 + 2 * Dex modifier (max 2).
It seems obvious then, that a "calculation" is something that sets your AC to a value plus other modifiers.
If something just adds on to (or increases) your AC, you continue to use whichever calculation is most favorable plus the new modifier.
Best Answer
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:
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:
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.