How difficult is it to build a device that can read credit card data from the mag-stripe...from scratch?
How do the magnetic grooves within the Mag Strip get converted to an audible signal - is there a name for this effect?


Induction.

You move a magnetic field relative to a conductor, you induce a current in that conductor. So, that's plain school physics here.


  Where would someone source a Tape Head such as the ones used by Square in the first version of their Card-Reader? Anywhere that they can be purchased online, or do I have to pull apart old Tape Decks?


Exactly where tape deck manufacturers source their tape heads. Online :)

No, seriously, there's giant electronic distributors out there, and they probably also have tape heads. If you need let's say a dozen tape heads, might still be cheaper to pull them out of junkyard stereos.


  How is the auditory signal transformed from a card swipe to digits? 


Let's get this straight: the audible signal is just a conversion of the electrical signal to an audible one. You wouldn't normally process the audio signal, but the electrical signal directly.


  Is there a formula that is applied to parts of the waveform?


Yes. Would be impossible without. 

Really, it's just different magnetizations. If you draw the strip across a coil, you can measure and amplify the induced current. From that waveform, you practically directly get the bits. It's pretty low-tech.


  Is there any software available?


Probably. Usually, it would be done in simple hardware (all this technology is pretty old by now, more than 40 years!). But yeah, why not: You already noticed that you can amplify the signal and make it audible. So, maybe just feed a sound card with it. From there on, it's just a few steps of digital signal processing that you'd need to design in your PC.


  (I can't imagine there would be any legitimate third-party non-enterprise entities fofering this functionality...)


what? Magnet stripe cards are used everywhere – in banking, in hotel room locks, in telephone cards, in public transport, in parking lots, in hospital access control systems, in time card systems …

So, that's a very ubiquitous application. There's a lot of card reader modules that you can simply go and buy. In single quantities, or thousands. Online. If you haven't found one, you haven't looked very intensely.


  Would all of this be easier to do by just buying a generic Credit Card reader from Ebay? 


Counterquestion: How competent are you at analog and digital signal processing, and how competent are you at buying?

(also: don't buy stuff at ebay. You've been warned.)


  Does every card-read manufacturer have their own card-audio-signal transformation software bundled into their device somewhere?


That audio is really stuck in your head. There's no audio unless you convert your signal into audio. And there's no reason to do that.

But yeah, if you build a card reader, you take something that converts a changing magnetic field signal in to a current or voltage signal (something like a tape head), you take something that processes that current or voltage signal and converts it to a stream of bits, and then you take that bit sequence, and error-correct it and convert it to the "meaning" it has. That's your job as a card reader designer. You go and pick the components that do that for you, configure/design/program and test them, and integrate them into a product.