Lachias mines the crystals and employs magic craftsmen to produce high quality products, which are then sold through the Golden Rose. However, because it's seen as gauche for high class nobility to directly involve themselves with merchants and trade (that's for low class nobles and commoners), she asks Levory to keep it a secret that the Lachias family is directly dealing with the Golden Rose merchants. The finest products were sent to the most influential nobles in society, (Influencers of the time lmao) such as the royal family, as gifts and as a marketing tactic. Then, the Duchess of Lachias arrives in the capital and throws lavish parties full of Golden Rose products. That the recently broke Lachias duchy can throw these exclusive parties full of hard to get products lets everyone in society know that the source of the ice crystals is the Lachias duchy, and that they'll soon be filthy rich from selling the Lachias catalogue through the Golden Rose. This earns Marquis Cameron's ire, as he previously had a monopoly on the mana stone trade. The royal family then invites Ran to the palace.
The comment about the ice crystal mine being her dowry comes from Ran being the current head of the Lachias Duchy; whoever marries her while she's the head will be in charge of the Lachias Duchy (Which is why in a previous chapter she said that she wouldn't marry while she was acting as the head, remember?). It makes her a very attractive marriage partner. And remember, the nobles don't believe that she'll really step down from her position as head once her younger brother comes of age. So in the minds of some of the aristocracy, she's currently 'the greedy lowborn duchess who usurped her highborn brother, the rightful heir'.
Ran's conversation with Yustav about taxes:
Because of the barren land and their duty as guardians, the Lachia Duchy currently enjoys low taxes. However, as soon as the Emperor finds out that they have a valuable mana stone mine, he'll impose large taxes. And because he's currently greedy, the taxes will probably be unreasonably large.
If they kept quiet about the mine they would avoid the tax increase, but then they won't be able to sell the crystals. If they refuse the taxes they'll probably go to war, and they still won't be able to sell crystals. So Ran and Yustav come up with a plan to make sure that the taxes the Emperor imposes are reasonable. We'll probably see it in the next couple of chapters.
Actually, I find this all pretty interesting, and the only place where I see a gap of logic from the author is what time the dressmaker came.. If the tailor came right before her visit to the palace, does that mean she was throwing those lavish high society parties to advertise her products in last season's clothes? Or was the scene with the tailor a flashback?