Technologically, fax machine is far superior than a printing press: its faster, shinier, and more efficient; however, the printing press is needed for there to be a fax machine.
Not really. There is absolutely no component created for the printing press that is needed for making a fax machine.
Would there be fax machines without printing presses?
No.Yes. The initial use of fax machines in the 19th century was for the quick transmission of images and signatures, not multiple copies of a single document. The most common use of the Pantelegraph (the first commercially successful facsimile machine) was for signature verification in banking transactions. The necessary precursor of the fax machine is not the printing press, but of the telegraph.
Yes and no. It is conceivable that the printing part of those devices could have been invented without the invention of the printing press so as to duplicate originals. Of course any device that abstracts the idea of a character and is able to turn a character into an image does depending upon the first mechanical implementation of that idea, the printing press, but fax machines don't print characters, they print images and thus do not depend upon the development of movable type as a necessary precursor.