1) What do we want to show of dwarf culture? How homogeneous do we want the clans to be?
2) What is the relationship between Mim and the rest of the dwarves?
3) How do the Sindar encounter the dwarves?
1) This is a more complex question than I first realized.
[DIGRESSION, BUT WITH A POINT! I PROMISE!]
Considering that Khuzdul was given by Aule to the dwarves and therefore is in some sense sacred, I've long suspected that the language would change much less over the ages than spoken languages, though bastardized slang and mannish vocabulary surely would enter in eventually. Maybe at some point the language becomes "frozen", like Latin did in the early Middle Ages, and becomes a language of court, law, and scholarship more than a spoken language. But I think that process would be very, very slow. Much slower than human languages, and maybe even as slow as elven. [/DIGRESSION]
ANYWAY, because I tend to think of Tolkien cultures in a primarily linguistic lens, I realized I had been assuming that this shared language would create a rather homogeneous culture among dwarrow, but I'm not sure the text supports that -- the Sindar have their differences, as do the Quenya-speaking races, after all. Still, I do think that dwarrow culture would be highly interconnected and lean towards insularity. They're always going to be a minority in M.E., given their low birthrates and dearth of females, and we have plenty of textual evidence for dwarves having an "us-vs-them" mindset. I do think fashion varies from kingdom to kingdom, and there should be some visual distinctions between the Families (though that would probably fade as years went by, due to intermarriage). I think we can show dwarves having vast trading networks that connect their different settlements and kingdoms long before men do, and cultural blending tends to follow trade.
We can't, however, have each dwarven city or kingdom look identical: it's boring and lazy, not to mention confusing for the audience if we ever plan to show more than one dwarrow place. An easy (-ish?) lead to follow here would be the Fathers: the craft best-loved by each Father might give visual clues to their descendants' surroundings: if the Father of the Firebeards, for example, was a glassblower (too on the nose?), than circular shapes and a blending of color and smooth surfaces might be very popular, but things might tend to be more compact or dwarf-sized, whereas if Durin was a stonecarver or mason, well, then it makes sense that his great kingdom has some truly astonishing architecture, but it might be a bit bare besides.
This is too say that I suspect the mores, attitudes, and mindsets of dwarves across Arda are probably rather similar, with the exception perhaps of the petty dwarves, but that their fashion and architecture could vary quite largely, if we wanted.
2) I don't have that much to say here, other than that I agree with Haakon that the dwarves should kick Mim out. I would not have his exile be "the first iniquity the elves inflicted upon the dwarves" but rather come from his own people. Why, though? Is he a weirdo outsider they're happy to see go, or are they assuming he'll gather up his stuff and move back to town? Is it his king who kicks him out to give Thingol the caverns, or is his presence there a surprise to the dwarrow workmen sent to prepare the halls for the elven king?
3) I have the idea of a Sindar scouting party, maybe with Mablung?, hearing the sounds of quarrying, and coming to investigate. It could be a play on the theme of hearing music/singing to new peoples, and establishes dwarves early as hardy, crafty folks, especially since the Sindar don't mine yet.
EDITED IMMEDIATELY TO HAVE: Is it obvious I have thoughts about dwarves? I have thoughts about dwarves.