Is “on” a correct preposition for “dog walk”?

Yes.

Yes. It is on \[an action\], in this case, on a dog walk, or it could be simplified to on a walk.

Absolutely.

Yes, is correct the same way you could say “…while walking our dog” both way are ok

Yes.

Yeah, it makes sense, particularly in American English. In British, I'd say "while walking our dog" sounds more natural.

Yes.

The phrase is "on a walk" and dog is an adjective describing the type of walk

Yes, on a walk, on a journey.

Yes, since it's also correct for "on our walk".

Personally, I would say “on our walk with our dog”, or “while walking our dog”. I wouldn’t use “during” or the term “dog walk” at all, they sound weird to me. -Native English speaker (US) and professional yapper.

Yes