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.
Composite Plating and Heavy Plating are intended to count as armor.
Keith Baker, original creator of Eberron and lead designer for the Wayfinder's Guide to Eberron, addresses this question (and many others) in the WGtE FAQ on his website:
Does the Warforged Heavy Plating Integrated Protection work with the Fighter’s Defense Fighting Style? Unclear on if (armor) means you’re “wearing armor” or not.
The intent is while you are using darkwood core you are not considered to be wearing armor for purposes of game effects such as Unarmored Defense, while you are considered to be wearing armor when you’re using one of the plating modes. However, as written Integrated Protection doesn’t allow you to benefit from the Defense Fighting Style. This won’t be changed until people have a chance to provide feedback on the race mechanics as they stand.
Baker points out that the feature doesn't specifically clarify that Composite Plating or Heavy Plating count as armor - but the parenthetical after each one reflects the intent, which is that Darkwood Core counts as being unarmored and that the other two options count as being armored. This is relevant for features like the Defense fighting style (which requires you to be armored), or the monk's Unarmored Movement feature (which, as the name implies, requires that you be unarmored).
In another response, he also says:
When a warforged is using the darkwood core (unarmored) mode and have light armor proficiency, can they choose whether they are considered to be armored or unarmored?
No. You are always considered to be unarmored when using darkwood core, and always considered to be armored when using plating.
So you can choose which of the three modes to use (depending on what armor you're proficient in), but you can't choose whether that mode counts as being armored.
...but as currently written, they don't count specifically as medium or heavy armor.
Baker also answers a related question about what kind of armor the Integrated Protection feature counts as:
Composite Plating and Heavy Plating both say “Armored” but do they count as medium and heavy armor respectively? Specifically does a warforged Barbarian in Heavy Plating mode lack access to the benefits of rage?
As written, the current version simply specifies that plating counts as armor, not that it counts specifically as heavy or medium armor. So in this version, a warforged barbarian can rage while in the heavy plating mode… though they will have to acquire the heavy armor proficiency before they can use heavy plating.
This may change in the future, but as of right now, a warforged barbarian can gain the benefits of Rage regardless of what defensive mode they're in. However, they can't benefit from features that forbid any armor at all while using Composite Plating or Heavy Plating.
Best Answer
Light vs Medium is mostly just a build choice; Heavy armor is somewhat more powerful.
Medium and Light armor both max out at the same point, AC 17. In Medium armor, that's achieved by the relatively simple means of having a +2 Dex bonus and wearing Half Plate armor (expensive but not ridiculous); in Light that's achieved by wearing studded leather and getting the maximum score of 20 in Dexterity for a +5 bonus, which may require many levels depending on your race and starting ability score. For the purpose of this discussion, I'm ignoring magic armor, which applies to all armor types equally, and effects that would allow a character to break the ability score maximum, which are typically what you might call "end-game content".
Medium armor is potentially "more powerful" than Light armor because it requires a smaller investment of resources to get to that maximum AC of 17. This makes "full AC" more readily available to a wider array of characters -- everyone has gold, and a 14 in Dexterity is available to virtually any character if they really want it.
For races that lack a racial bonus to Dexterity, such as the Hobgoblin, Medium proficiency is a slightly more valuable armor proficiency than Light. Without a racial bonus to Dex, reaching that maximum score of 20 will be more costly and time consuming, and if your Dex score is below 14, Medium armor allows for a better AC than Light armor does, both overall and dollar-for-dollar.
That said, Medium proficiency is already widely available. There are many classes that start with it, and if you're allowed to use multiclassing, a single level dip into the vast majority of classes will grant Medium proficiency if you really want it. In general, adding Medium proficiency instead of Light at the race level only really benefits a handful of classes, such as wizards, sorcerers, most warlocks, and some bards.
By contrast, Heavy proficiency is definitely more powerful than the others. While acquiring a suit of full plate is fairly expensive, the AC 18 that it grants is beyond what's available from any Medium or Light armor, and it's available to virtually any build. There is a minor price in speed if you aren't pretty strong or a dwarf, but assuming you're willing to deal with that, there's not much of a downside to wearing Heavy armor.
It's not accidental that Heavy proficiency is relatively hard to get. It's available to only a few classes to begin with, none of which offer it as a multiclassing proficiency. Gaining Heavy proficiency after level 1 requires a valuable feat slot, or the use of a non-obvious backdoor by multiclassing into one of a very small number of specific class options that will grant Heavy proficiency through a class feature (most notably, multiclassing into Cleric with an appropriate domain choice). I feel that the relative rarity of Heavy proficiency and difficulty of acquiring it is the developers telling us that Heavy proficiency is a meaningfully powerful class feature, and that in itself is a good enough reason not to simply hand it out to an entire species.