I haven't quite absorbed all of this yet in it's entirety, but I'd like to make a start, because I feel this is a fantastic way to broaden the game, and also the game's audience base.
Firstly:
I would love to see this incorporated into migrant waves, so that the social classes that your Fortress most appeals to, will be attracted to it.
To give some simplified examples of this: If you have a primitive economy, where you mostly produce and sell raw resources, you should attract large amounts of unskilled laborers, because they're seeking jobs--any jobs at all.
If you make and sell a lot of toys, you should be attracting younger couples, and dwarfs with children, along with skilled artisans.
Making weapons and armour should attract grim veterans and mercenary-types. If you then export those arms, you may also attract less savoury characters interested in arms-dealing, but you would also be increasing the strategic value of your Fortress, and that should alter the landscape, too.
Wealthy Fortresses would naturally attract the rich, and those seeking wealth. If you make the streets safe to walk down at night, you'll attract permanent merchants and business-folk, and they'll bring their families.
If it's not that safe, internally, you'll still attract tourists, and those on temporary business, but you're going to also attract crime.
If it's not that safe, externally (in other words, you're attracting seiges and titans, and not dealing with them quickly), you may start to attract a larger military presence, and that outside military may not be under your direct authority, so it may function like caravan guards. In practice, these military detachments may cause you social problems that reach across social class boundaries--could be something like having a whole squad of Hammerers show up out of the blue, that you have even less direct control over.
They would be a well-equipped veteran or elite squad, so they could help you defend your Fortress, to a point, but they might only show up if your Fortress is considered strategically worth it, they might not be worth the internal social upset (and you ofcourse would have to feed, house, and pay them), and they might be a one-time deal.
Secondly:
I think divisions of social castes could allow you to create conditions where your Nobles become real assets, rather than annoyances.
If you use a historical model, then the nobility was responsible for protecting their lands, as much as ruling them.
That was the arrangement that they made with the peasants--the peasants would work to provide food, and they would submit to the rule of the noble classes, but in trade, the Nobles were (atleast in theory) required to produce elite warriors to protect those lands and the people on them, and these elite warriors were each expected to spend atleast a full third of their time training, and an enormous amount of wealth on armour, weapons, atleast one carefully bred and well-trained warhorse, and to provide atleast one fighting squire, and then hire or train from several to as many as a few hundred men-at-arms, to back them up, in times of war.
There may be a sort of social "economy", where, once you fill all of a Noble's needs (furnished room, furnished office, etc.), you are then presented with some kind of a Noble "wishlist", and if you fill that, the Noble starts fulfilling obligations to you, in return. They may start by purchasing their own weapons and armour, and move on to training in martial skills, and then to equipping, educating, and training a small "household" of other dwarfs, and their own children.
If you create conditions that are favorable to rich, noble, elite dwarfs, those dwarfs should be atleast partly responsible for the defense of that Fortress, so you should be getting something out of the arrangement.
Thirdly:
A person's social caste should realistically be as important as their species, in many ways. It wouldn't decide their fate, but it would greatly influence where they start in the world, who their parents were, how they view money, and how much money was spent on raising them, what people and what skills they were exposed to, their feelings on religion, politics, and outsiders.
It's more than just social status, it would strongly influence every aspect of their everyday life--food, bathing, prayer, romantic relationships, entertainment--and the reactions of everyone they meet.
To start with, I'd personally like to see it be very difficult for someone who isn't a member of the "rich noble warrior/knight" caste to train in the following skills:
Armour User
Shield User
Fighter
Dodger
Sword User
Pike User
Bow User
Leader
Negociator
Reader
Weaponsmith
Armourer
Alchemist
Record Keeper
Military Tactics
I think that peasant castes should learn these skills at maybe 1/4th the normal speed, until they reach Professional level, at which point they could be considered professional soldiers, and would then progress normally.
Conversely, those born into other castes could also learn skills at different rates, to simulate limited or specialized training, access to or denial of resources, information, and money, how exposed the caste would be to outsiders, etc.
There would be tools that peasants would naturally have access to, ofcourse, and be able to train on and use, and many tools can easily be weaponised, but chopping down a tree shouldn't be computed in the same way as dueling with a goblin.
A peasant and a knight would view each of those tasks from very different perspectives.
As far as concerns about social climbing: I think that should be modifiable under the Entity tag. Some civilizations are going to be more flexible than others. Some allow peasants to own weapons for hunting and self-defense, while others don't allow any weapons at all--Japanese martial arts were very directly influenced by social castes, and the need for those castes to interact.
Aikido and Karate, for instance, were both invented as a result of laws about who could and couldn't use weapons.
The crossbow was viewed as a coward's weapon, because any peasant could hide in a bush and kill an armoured knight with one. Swords became iconic with war, in part because it specifically was not a tool--it was the opposite of a plowshare.
Farmers weren't always on the lowest rung, either, and the wealthy weren't always on top. To go back to Japan, the farmers were right below Samurai, socially, and it was the wealthy merchants who were actually the lowest of the low, forbidden from spending their immense wealth on basically anything other than food and entertainment.
Entites could have "ethics" dictating:
Which castes can own, purchase, or even handle, which items--weapons would be obvious here, and a member of the wrong caste just touching a sword could be grounds for execution.
Which castes could perform certain tasks (India had a caste system that designated certain populations as the only ones allowed to touch corpses, in order to bury the dead and prepair for funerals, for instance, and descendants of that caste still suffer persecution today.), and learn which skills, including who could learn to read and write, who could be a metalsmith, who could be a doctor, who could be a priest, etc.
Which castes could eat certain foods and drink certain beverages (eating hearts of palm was punishable by death unless you were royalty in Polynesian culture, and hot chocolate was, famously, a drink fit for an Aztec emperor, and only the emperor),
Who could wear certain clothing, and certain weaves of cloth (There's a special weave of cloth that only the royalty of Ghana is permitted to wear). Clothing styles would ofcourse say a lot about social caste, but this may be more stylized in some cultures than others,
Where you could live, where you could worship, where you could be buried, who could kill you and get away with it, and your supposed status in the afterlife.
Dwarfs are not humans though, and there's no reason to assume that such a strongly craft and tool oriented culture wouldn't naturally gravitate towards a government of meritocracy, where the most skilled dwarf rules, so they may have no strict policy against social mobility.
Ofcourse, in modern society, it's not easy or cheap to become a neurosurgeon, or an astronaut, and both require a lot of intense training and study, so there's no reason to assume that social mobility would still be fast or certain.
High (er...low?) dwarf society families may horde libraries, import foreign tutors, enforce strict Guild requirements (and outright bribe the Guilds), or even create secret Greek Mystery and Mason-type societies--mystically steeped shadow guilds-behind-the-Guilds, designed to obscure technical know-how, and make sure "only the right dwarfs" make it to the top of their professions.
There's also the question of exactly how many social strata there are in a single society, and where each caste comes from. Is a peasant considered to be a slave, a serf, a yeoman, a freeholder, a citizen-soldier? Does a peasant have rights? Does a slave have rights? Which one is more likely to climb, socially, and why? Are these castes "the will of the gods", or are they just practical means of making sure that society keeps on ticking?
Thirdly:
I feel that migrants coming into your Fortress would be looking for a better life for themselves.
It would be great if you could fulfill their hopes or shatter their dreams by deciding to enforce the traditional status quo, significantly alter it, or even develope a significantly different society than that of the Mountainhome--you could even decide to break off from the Mountainhome, in a glorious rebellion.
I think it would be important for the game to have a more robust legal system in place, and also of religion, and then tie them both into the social order (Where do priests fit in? Where do the police fit in?), the economic environment (including Guilds, which should be very important and powerful entities of society), and politics, and tie all of them together; so that you can't tug--atleast not without employing a lot of political delicacy--on any one of these "strings", without pulling on all of the others at the same time.
Once that's reached a point where it's a more holistic web, you could then figure out how every person fits on that web, what everyone's checks and balances are, and how they all connect, from the king on down.