Learn Finnish: Paljonko on kymmenen miinus neljä?

Both mean "how much" and are interchangeable here. Paljonko attaches the question particle -ko directly to the word paljon (much). It is very common and natural in spoken Finnish when asking for a specific amount, like a math result or a price.

Why are the numbers kymmenen and neljä in their basic nominative form? Shouldn't amounts often involve the partitive case?

While numbers usually trigger the partitive case for the nouns that follow them, the numbers themselves in a simple math equation stay in the basic nominative form. You are treating the numbers as abstract concepts rather than counting physical objects.

Is the word for "minus" really just miinus?

Yes! It is an international term adapted to Finnish spelling. If you wanted to add instead of subtract, you would use the word plus, making it kymmenen plus neljä.

Could I drop the verb on and just ask Paljonko kymmenen miinus neljä in casual speech?

No, you generally still need the verb on (is) to make it a complete sentence. Even in casual spoken Finnish, you include the verb to link the question word to the math problem.