You seem to have a few different concepts in there...
But mutations are always completely random and human beings have no control over it.
Aside from the fact that mutations are not completely random (not always, at least), it is not true that humans have no control over them. Imagine you grow plants, and you start to cross those plants that have a larger stalk, allowing them to be more resistent to wind: you are effectively selecting a certain mutation. Surely, you cannot choose which specific sequence you want to mutate, but still you are enriching your population of plants with a specific mutation.
Nowadays of course you can specifically mutate the genome in the lab, but that is a different matter.
Would it have been possible to domesticate dogs from wolves, if there would have been no mutations in wolves to begin with?
Note that artificial selection has been used to generate the different breeds of dogs, by breeding animals with specific characters.
However, domestication is a different matter. Wolves and dogs are the same species (Canis lupus) and domestication is not strictly dependent on selective breeding. You can domesticate an adult animal without inducing mutations in its DNA. Probably epigenetics plays an important role there, and epigenetic patterns could possibly be transmitted to the offspring.
Of course, then you will use selective breeding to expand the domesticated population, and to select those traits you are interersted into. With artificial selection you are just selecting the mutations you want, the way you generate them is irrelevant to the matter: for instance you can irradiate plants to enhance mutation rate, and consequently the appearance of new favorable traits.