Can you set speakers to produce any frequencies that you wish
Not any – I don't know the general range, but it definitely depends on the quality of the speakers, as well as your sound card (the "DAC", I believe). They're optimized for voice and music, and don't need to care about the rest. The same applies in the other direction (recording) as well.
However, it's doable even with a fairly limited frequency range – just look at good old dial-up modems (which originally used an acoustic coupler directly to a telephone handset).
As far as I know, they don't just use a different frequency for every byte, but encode individual bits. (Just like electrical communications don't use a different voltage for every byte, either...)
(As another example, recently various products have started using ultrasound to transmit secret keys and other pairing information, e.g. when connecting a phone app to a camera and such.)
As mentioned in the comments, you could start with the DTMF standard which can encode 16 different keypresses.