Dwarf Trainers for an organization?
One point for skilled but common in dwarf society (Miners), two for specialized units (Rangers), four for true expertise (Ironbreakers), eight for hero-level instruction. Would require pay on top of favour expenditure.
Expert assistance to establish a factory for turning saltpeter into gunpowder?
Five points.
Personally tutor Mathilde in any field not considered forbidden to non-Dwarves?
One point for skilled but common in dwarf society (Miners), two for specialized units (Rangers), three for true expertise (Ironbreakers), six for hero-level instruction, ten for contenders for most knowledgeable in their subject in the known world.
Or in Khazalid?
Will happen automatically if Mathilde spends much longer among Dwarves.
Let Mathilde move into a dwarf hold and live there?
If you promise to refrain from spellcasting inside the Hold, and would be willing to live in the bunkhouses, your rep would be enough.
While conducting magical research?
Four points, and you'd have to work out a way to have a laboratory far from the populated parts of the hold. Does not apply to K8P.
Send a force to assist in a campaign against someone they already hate
If they've outstanding grudges, and you're in a position of power in the campaign, and they've got the forces to spare, your rep would be enough.
Or that they don't care about?
Impossible.
Provide her with/sell her enchanted equipment of their make
No limit; if very large amounts are spent, her rep would give her access to Kragg or Thorek.
Or mundane but high-quality equipment, such as pistols?
Your rep means they'll sell them to you, or you can spend four points to commission something.
Normally
Or have an expert collaborate on an enchantment project with her?
Would need to be arranged personally with the expert, but your rep would get you an introduction and their ear.
Help Mathilde dig or renovate an underground base somewhere
Two points to dig, five to renovate it so that it'll last at least a thousand years. It would cost more to renovate it to a lesser standard.
In total secrecy?
As above, and you provide the ale.
Assist in construction of some key project, such as a bridge, road, or fortification?
If you mean consultants, same as the trainer scale.
Or just do the whole thing?
Would need actual payment, and the cost would be commensurate with the craftsdwarfship, but your rep would open the door to commissioning the work.
Share maps, information, and general intelligence on the things that dwarves know about events in the world but Mathilde doesn't?
Too vague to say. If you mean skaven, say skaven.
Provide her with letters of introduction testifying to her skill and trustworthiness?
From Belegar, free. Two points from non-KaK Kings.
Or a temporary companion/aide in diplomatic matters?
As per the trainer scale.
How much rep for political asylum if the need arise?
Depends who from, and for what. Varies too much to say.
And how much for a helicopter ride?
If you just mean a joyride, you can get it from your rep. If you want a flying taxi, a point for anywhere in the general area, two for anywhere friendly in the Old World.
Purchasing books for her library?
Default and extensive from rep alone, 1 point for rare or very rare, 2 for extremely rare.
