Rules as written: No limit
Armors limit your Dex bonus to AC. Deepwardens do not have Dex bonuses to AC, they have Con bonuses to AC. The Con bonus replaces the Dex bonus (and so you would not get it in situations where you lose your Dex bonus to AC), but it still is not a Dex bonus and thus the limits of the armor do not apply. And, indeed, there’s no reason why an armor’s weight or inflexibility would prevent you from benefiting from your resilience and connection to the earth.
Intent: Unknown, and unknowable
As far as what the authors intended, that’s anyone’s guess. The author never wrote anything on the subject aside from the original publication, so we have no evidence one way or the other.
Precedent: No limit
As for precedence, generally speaking the special rules for one ability score do not apply when you swap to another ability score. There are instances where you can replace your Strength to damage with Dexterity to damage, but RAW you do not get ½Dexterity to damage on a light weapon or 1½Dexterity to damage on a weapon wielded two-handed, unless it explicitly says so. (Dragon vol. 221 has the Corsair which explicitly halves Dexterity to damage with light weapons, for example)
FAQ: Limit
However, there is an FAQ entry on the question. The FAQ, it must be very clearly stated, is not errata. It is supposed to only explain the rules that already exist. If its statements contradict the actual rules, officially, the actual rules take precedence every single time. And the FAQ has been wrong many times, in many cases quite blatantly so. Finally, the FAQ is not (usually) written by the original authors of the book, so it does not correspond to the author’s intent, either.
All together, the FAQ is worth very little, and has a very poor reputation. In fact, it’s so bad that when this poster in the thread Squera linked learned that the FAQ said the Con bonus was limited, it proved to him that it actually wasn’t!
At any rate, the FAQ has this to say:
Does the deepwarden’s Stone Warden ability (RS 105) still have a maximum Dexterity bonus to his Armor Class, and does that maximum still apply to his Constitution?
The maximum Dexterity bonus should be treated as the maximum ability bonus given by the armor, so if you were playing a deepwarden wearing full plate, you would only add 1 to your Armor Class from your Constitution.
Conclusion: Rule it based on what improves your game most
Personally, I think it makes almost no sense that restrictive armor would inhibit a Con bonus to AC, I think that deepwardens make sense in heavy armor and shouldn’t be penalized for wearing it, and I don’t think the deepwarden or the AC they can get is overpowered or problematic, so I would not limit it. You may answer any or all of those questions differently. Ultimately, remember than AC is a limited thing, and lots of nasty things ignore AC; having a lot of it only does so much for you.
If you would like a compromise, here’s a suggestion. Purely houserule with no basis in the rules, either deepwarden’s or elsewhere in precedence, but it might be a good solution:
The deepwarden’s Constitution bonus to AC is limited by her armor’s maximum Dexterity bonus, just as the Dexterity bonus it replaces would be. However, the Constitution bonus is not limited for the purposes of AC against touch attacks. For example, if a deepwarden with 18 Constitution (+4) wears Chainmail (+5 AC, +2 maximum Dexterity bonus), and has no other bonuses to AC, his AC is 17 (+5 armor, +2 Consitution), but his touch AC is 14 (+4 Constitution). A deepwarden’s touch AC may not exceed her regular AC.
I don’t think this is necessary, but it does prevent the high-end AC that you might get from armor+Con, while still giving a deepwarden a good bonus for their class feature.
Remember, a class feature is supposed to make you better. Yes, Con, not limited by max Dex, is better than what the deepwarden had before, but that isn’t necessarily a problem. It’s only a problem if it’s “too much better,” and I really don’t think it is.
The Barbarian is Flat Footed
Under the initiative rules, the SRD says this (emphasis mine, also in PHB on page 137):
At the start of a battle, before you have had a chance to act
(specifically, before your first regular turn in the initiative
order), you are flat-footed. You can’t use your Dexterity bonus to AC
(if any) while flat-footed. Barbarians and rogues have the uncanny
dodge extraordinary ability, which allows them to avoid losing their
Dexterity bonus to AC due to being flat-footed.
Nerveskitter says this:
You cast this spell when you and your party roll for initiative.
Unlike other immediate actions, you can cast this spell while
flat-footed. You enhance the subject's reactions in combat, granting
it a +5 bonus on its initiative check for the current encounter. If
the subject does not make an initiative check within 1 round, this
spell has no effect.
Ferocity says this:
He can enter this state as an immediate action, even when flat-footed
at the start of combat, so he may apply the enhanced Dexterity
modifier to his initiative check.
Both of these abilities let you act as an immediate action even while flat footed, before your regular turn in the initiative order. Neither are your regular action in the initiative order, as they're letting you act out of turn. Thus, neither of them prevent you from being flat footed just by using them.
They might prevent you from being flat footed by making your initiative high enough that you get to act first in the normal order, of course.
Best Answer
You’re running afoul of some major name conflicts here: there are multiple different things that are called “dodge.”
So, three instances of the word “dodge” that are not actually the same:
Dodge bonuses. This is just a type of bonus, almost-always to AC. For clarity, the rest of the answer will refer to these as “dodge-type bonuses.” Dodge-type bonuses stack with other dodge-type bonuses (unlike most types of bonuses), and you lose dodge-type bonuses to AC any time you would lose your Dexterity bonus to AC.
The Dodge feat. This is a feat that gives a dodge-type bonus to AC, but only against one chosen target.
The Uncanny Dodge class feature. This is a class feature that allows one to keep one’s Dexterity bonus to AC in more situations. Since you don’t lose your Dexterity bonus to AC, you also don’t lose any dodge-type bonuses to AC that you may have (since you usually lose those when you lose your Dexterity bonus to AC). However, the Uncanny Dodge class feature itself does not grant any bonus, dodge-type or otherwise, it just affects when you lose any bonuses you may or may not already have.
Note that the restriction of applying only to one target is a facet of the Dodge feat; it does not apply to any other source of dodge-type bonuses, nor does it have anything at all to do with Uncanny Dodge.
So, for example, the dwarven dodge-type bonus to AC against giants applies to any and all giants, all at once. You do not have to pick one. If a dwarf has the Dodge feat, he can apply the Dodge feat’s dodge-type bonus against the attacks of any one creature. If that creature is a giant, the dwarf gets to stack his racial bonus alongside the bonus from the feat, for a total of +5. His dodge-type bonus against other giants remains the same +4.
If he does not have Uncanny Dodge, he loses all of these dodge-type bonuses when caught flat-footed or attacked by an invisible attacker, since he loses his Dexterity bonus to AC and dodge-type bonuses are tied the to Dexterity bonus. However, if he does have Uncanny Dodge, then he gets to keep these dodge-type bonuses in this case, because Uncanny Dodge lets him keep his Dexterity bonus.
So, for your example:
Dex-16 4th-level dwarf rogue with leather armor: basic armored AC of 15. Note that dwarves are Medium creatures, and therefore have a size bonus of +0, not +1. (Only Small creatures get a +1 size bonus to AC.)
Touch attacks ignore the leather armor, so against those his AC is 13.
Since he is a rogue with Uncanny Dodge, being flat-footed does not cost him his Dexterity bonus to AC, so his flat-footed AC is the same as his regular AC. It is still possible to lose one’s Dexterity bonus to AC with Uncanny Dodge, however (balancing is a big one); if that happens, his AC is 12.
He gains a +4 dodge-type bonus against any and all giants, so that includes our two hill giants. For this fight, against only giants, his AC is 19. Uncanny Dodge means he keeps those dodge bonuses even if caught flat-footed or the hill giants are invisible (but not if he is balancing or otherwise does lose Dexterity to AC; then his AC is back down to 12 since it takes out the dodge-type bonuses as well as the Dexterity bonus).
He may designate one of the hill giants for his Dodge feat. His AC against that hill giant only improves to 20. Against the other hill giant, his AC remains 19.
Flanking doesn’t affect AC. If the hill giants are flanking him, they get a +2 bonus to their attacks. If they are also rogues, they would also get to add their Sneak Attack damage to their attacks, as appropriate.