As you probably already reckoned from the comments, there are many alternative ways to express the English meaning of "how adverb do you verb". Which is more "natural" will depend on the local dialect and the register.
In the formal register the "correct" form employs the question word cuán, with an adverb following. There are a number of adjectives that, while having a regular adverbial derivation in -mente, are often used adverbially in their singular masculine adjectival form. These include rápido, despacio, lento, fuerte, etc. So for your example it would be either of the following:
¿Cuán rápido vas a caminar?
¿Cuán rápidamente vas a caminar?
Other less formal ways employ the phrases cómo de and qué tan:
¿Cómo de rápido vas a caminar?
¿Qué tan rápido vas a caminar?
If the adjective is not one of those that can be used adverbially, you will always have to add -mente. The rest will be the same.
You may in some cases use a phrase like con qué [noun] as you proposed (¿con qué rapidez...? etc.) but this sounds distinctively forced in most contexts that I can imagine. It may be the appropriate form in technical speech:
¿Con qué velocidad partió el proyectil?
Finally, your idea:
¿Cómo [adverb] [verb]?
This is ungrammatical and you shouldn't use it. It's a straight calque from English how [adverb] but Spanish doesn't work like that.