Learn Spanish (Latin America): ¿Qué tan afilado está ese cincel?

While cómo means 'how' in the sense of 'in what way', it cannot be followed directly by an adjective to ask about degree. In Latin America, to ask 'how [adjective]' you must use the phrase qué tan.

Does the word afilado change depending on the object we are talking about?

Yes. Even though it comes right after qué tan, afilado is still an adjective describing the noun, so it must agree in gender and number. Since cincel is masculine singular, it remains afilado. If you were asking about a sword (espada), you would ask ¿Qué tan afilada está?

Why does this sentence use está instead of es? Isn't sharpness a physical trait of a chisel?

We use está here because sharpness is viewed as a changeable condition, not an innate, permanent characteristic. A chisel gets dull with use and then gets sharpened again, so its current level of sharpness is a temporary state.

I have seen cómo de used to mean 'how [adjective]'. Could I use that here instead?

Yes, ¿Cómo de afilado está...? means the exact same thing, but it is primarily used in Spain. In Latin America, qué tan is the standard and most natural way to ask about degree or extent.