I am going to be quoting from Eberron materials from the “v.3.5 revised edition” of D&D quite liberally in this answer—because that is where Eberron originated, and it’s got over a dozen books for Eberron, while 5e has just Eberron: Rising from the Last War. There’s a lot more material to use, and I just know where it is better. I have searched through Eberron: Rising from the Last War for material relating to the precise requirements of the houses’ neutrality, but (as far as I can tell) there just isn’t much of anything on it to be found in that book.
Not technically citizens of any nation, the matriarchs and patriarchs of each house live in splendor within their enclaves and emporiums located throughout Khorvaire.
Your quote is referring to the various agreements that ended the War of the Mark, as well as the Korth Edicts some time thereafter, which together codify the modern meaning of “dragonmark house” (and also the persecution of aberrant marks).
Unfortunately—sort of—these agreements themselves have never been precisely written down in a sourcebook, only generally described. I say “sort of” because, of course, this leaves room for the DM to nail down the details in particular ways as best suits the campaign, which is a good thing, but it does make answering this question difficult.
Things we do know in a general way about these treaties and the houses’ neutrality:
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−1,500 [years ago] […] The dragonmarked houses launch the War of the Mark to end the threat of aberrant and mixed marks. The Twelve is established.
[…]
−1,005 [years ago] Galifar makes a deal with the dragonmarked houses, offering them neutral status in exchange for support in his campaign [to unite the Five Nations].
(Eberron Campaign Setting pg. 225)
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The agreements at the end of the War of the Mark formalized tenets and traditions that all the houses adhere to. These include rules concerning neutrality (all houses make a show of remaining neutral so they can operate in all nations), fair trade and practices, family naming conventions, and a ban on marriages between houses so that dragonmarks remain pure.
(Eberron Campaign Setting pg. 231)
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For a thousand years, the Korth Edicts prevented any member of a dragonmarked house from holding a grant of land and placed limits on the size of house enclaves and the armed forces garrisoned there. […] The edicts further specified that no member of the aristocracy of Galifar could be bound to a member of a dragonmarked house in marriage without one of the two giving up all heritage and rights. Since the houses did not own lands, the edicts dictated a system of rents to be paid to the crown in exchange for the territory the houses required for their needs.
(Dragonmarked pg. 10)
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With the hounds of war baying and the dragonmarked houses ascending, the Korth Edicts were quietly set aside during the Last War. Even now, many house enclaves maintain forces beyond those allowed by the edicts, and a number of houses have successfully claimed land and holdings of their own. One notable example is Stormhome in Aundair, which is for all intents and purposes a territory of House Lyrandar.
In the wake of the war, the status of the edicts remains uncertain. The Treaty of Thronehold called on the authority of the Korth Edicts when House Cannith was ordered to shut down the creation forges. Though Cannith acquiesced, the house was in chaos at the time, its leadership shattered on the Day of Mourning. Today, more and more dragonmarked nobles are holding to the opinion that the edicts were an agreement with the King of Galifar, not the rulers of the Five Nations. It remains to be seen whether the Thronehold nations can join together to enforce the terms of the edicts once more—or whether the growing economic and military power of the houses will allow them to dictate new terms to the nobility.
(Dragonmarked pg. 10)
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As an arm of the dragonmarked houses, the Twelve attempted to remain neutral during the Last War, but many of its wizards and artificers had national loyalties that were stronger than their ties to the institute.
(Eberron: Rising from the Last War pg. 74)
There are more such mentions, but these are the meat of what we have. I seem to recall (but cannot immediately find) statements that House Cannith at least somewhat favored Cyre in the Last War, for instance. Notoriously, Sasik d’Vadalis, brother of the leader of that house, is consort to Queen Aurala of Aundair—without relinquishing his standing in House Vadalis (but then, technically, without marrying the queen). House Kundarak—formerly Clan Kundarak—no longer votes in the Iron Council of the Mror Holds, but it has substantial political sway over the other clans that do. And so on. Each of these is seen as “pushing the envelope” in terms of house neutrality.
With that in mind, we can try to answer your questions.
Is this true for ordinary members of the house? Are they citizens and can be drafted into service?
This is unclear, but the example of the Twelve from Eberron: Rising from the Last War pg. 74 suggests that members of the houses can have, if nothing else, at least emotional allegiance to one nation or another.
However, consider that Galifar was a kingdom, that is, a feudal society. The concept of “nation-state” doesn’t really apply in feudal societies—allegiance is generally to a person, not the concept of a nation. Often, that allegiance isn’t even to the person in charge of the nation—e.g. the king—but to the person directly above you, who is sworn to the person above them, who is sworn to the person above them, and eventually someone is sworn to the monarch and that’s why you’ve got to listen to them. If someone in that chain defects, you were usually coming with (unless someone below them broke their own loyalty to remain loyal to the monarch).
This is relevant because the heads of the dragonmarked houses are each barons. The heads of the houses are certainly held to neutrality—per the Korth Edicts, they would have been sworn directly to the monarch before the Last War (this would be, in the real world, not unusual for barons, who often were directly sworn to a monarch and not to one of the dukes, counts, and so on who were above them). And then every member of the house would be sworn to the baron in charge of it (or one of the barons in charge of it, since sometimes the leadership gets split, e.g. modern House Cannith has three barons).
With the dissolution of the Kingdom of Galifar, the heads of the houses are no longer sworn to anybody—there no longer is any king or queen of Galifar, and they are legally barred from allying themselves with any other nation (so long as the Korth Edicts hold, anyway). Since everyone else in the house is sworn to them, that suggests that no one in the house has any proper allegiance—that is, responsibility—to any nation.
That would prevent any nation from drafting any member of a dragonmarked house into service. It probably wouldn’t, however, stop them from enlisting, if they felt some patriotic duty to do so. We know this happened to some of those working at the Twelve.
Can a nation draft a member of house Orien to run a lightning rail service, even if the matriach wants it to be shut down?
House Orien owns the lightning rail, so that’s definitely out—even if members of the House are citizens and drafted, the lightning rail isn’t their personal property and they have no right to dictate where it goes or when it runs.
In reality, nations that have particular needs for lightning rail service simply pay House Orien exorbitant amounts to offer custom service. This is expensive, but at least currently, trying to force them to do so is beyond the capability of any nation, particularly one that is hard-pressed by other needs (e.g. whatever they need the special service for).
This is true for most, if not all, of the dragonmarked houses—they’re about on par with actual nations for power and wealth. A nation trying to force them to do anything is going to have to go to war to do it—and the house will likely have many ready allies in the form of 1. the other dragonmarked houses, who don’t want to allow the precedent of a house being pushed around like that, and 2. the other nations, who don’t want to allow any other nation to get special treatment.
How common are people outside of houses who have the dragon mark? […] I could assume a secret affair or sex with a prostitute which creates a child, who never knew who his father is.
Depends what you mean by “the dragon mark”—but the answer ranges from “rare” to “extremely rare.”
Rare would be aberrant dragonmarks. These are unique dragonmarks, different from the thirteen “true” dragonmarks (Death, Detection, Finding, Handling, Healing, Hospitality, Making, Passage, Scribing, Sentinel, Shadow, Storm, Warding). They show up randomly, and are considered evil or corrupt by most of Khorvaire—the War of the Mark was largely about them and the dragonmarked houses’ attempts to eliminate them. It’s unclear if there’s anything actually wrong, or even different, about these marks, or if that’s just propaganda and any one of these marks could develop into a new dragonmarked house if allowed. They are rare, but when they do show up, it’s (basically?) never within an established dragonmarked house.
Extremely rare would be one of the “true” dragonmarks. For one of the twelve remaining dragonmarks to appear outside of the associated house, basically means the marked person has an ancestor who was an illegitimate, unrecognized child of the house, or an ancestor who was kicked out of the house. Most of the houses are very permissive about recognizing bastards to prevent exactly this kind of thing, and kicking a member of the family out of the house is rare in the extreme. One way or the other, though, “a character with a dragonmark can always find a connection to a dragonmarked house somewhere in the roots of his family tree,” (Dragonmarked pg. 7).
(Outright impossible—barring divine DM intervention—would be the re-appearance of the Mark of Death. That dragonmark has been thoroughly stamped out, and no one is going to discover “a connection to [House Vol] somewhere in the roots of his family tree.”)
Sometimes it skips a generation or just shows up very late.
Correct; Dragonmarked states that “approximately half the children born to dragonmarked parents eventually develop dragonmarks of their own,” (pg. 7).
