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.
RAW: Somewhere between levels 5 and 11
Six levels is a lot of variance, so let's see what we can discern. Looking over the expected results from the treasure tables in chapter 7 of the dmg and the "starting gear" table for adventures not starting at level one, a character (or party, depending on loot-sharing) should be able to obtain plate mail by around level 7; A level 5 party may be able to afford this by pooling their resources.
The tables are all somewhat imprecise, but we know that at level 5, you start with about 600GP and at level 11, you start with about 5600GP. Note that ~600GP is consistent with expected loot from CR-appropriate monsters during levels 1-4.
IF we assume that the scaling is actually linear and not tiered (meaning that a smooth-ish line can be drawn from each level's expected loot, rather than a 5th level character having 500 more gp than a 4th level character) and we use the two points mentioned above (level 5 with 600 gp and level 11 with 5600 gp), then we can find a distribution. (though, if such a graph were to exist, it would probably not be linear, but I digress)
5000GP over 6 levels is ~850 gp per level.
That gives us:
level 5 with 600GP
level 6 with 1450GP (just shy of full plate)
level 7 with 2300GP
level 8 with 3150GP
level 9 with 4000GP
level 10 with 4850GP
level 11 with 5700GP (close enough)
from which we can derive.....
Around level 7 for an individual
Around level 5 for a party
Best Answer
As far as the rules are concerned, you've already quoted everything that is relevant here:
There is no caveat there about "while moving" or "except when standing still", so yes, the penalty still applies when the wearer is standing still.
You are (as always) free to rule otherwise in your games, or if you're a player, ask the GM to rule otherwise. Of particular note here is the description of Dexterity checks:
So if you're standing still, the check might not be a Dexterity check. The description of the Stealth skill doesn't contradict this, either:
Nothing here suggests that standing still should be modeled as a Stealth check, so it could be worth asking your GM (or ruling, if you are a GM), for some other kind of check to model this. The section (I won't quote it all here) on using skills with different ability checks (PHB pg. 175) may be useful for supporting this.