@fraq has dropped a bunch of concepts here that may be confusing for people not familiar with communications systems. So I will try, with @fraq's permission, to expand a bit further. Sure. I like these topics
As @fraq said, that elephant thingy is related to mechanical waves, like sound, not electromagnetic waves. You can always use transducer to convert one type of signal into another tho. Both are waves, but one moves some physical thing (air for sound) and the others "moves" electric and magnetic fields.
Your best option to transmit signals over long distances is to use longwaves (see also VLF). Or short waves bouncing your signal into the ionosphere... but in that case it will be a kindof point to point link. The best you can do is probably trying a Moon Bounce.
FM (and PM) and AM are analogue modulation techniques. You can decide to use some analogue representation of a digital signal using these modulation, but it is better to use a digital modulation. Their counterparts are called FSK and ASK, being those the simplest digital modulation you could use. But probably you will find more often a QAM with some complex symbol constellation. Anyway, at the end, you will get an analogue signal at a certain frequency (that is how nature works ).
Finally, the Nyquist theorem is related to sampling, not even to communications. Sampling is the process of converting an analogue signal into a digital signal. For doing that you have to take samples of the continuous analogue signal every certain time (the sample frequency/rate). The Nyquist theorem basically tells you that you need to sample as fast as twice the higher frequency in your analogue signal, to be able to capture all the information it contains.
If you sample at a lower frequency, you may need an antialiasing filter (that will limit the higher frequency in your analogue signal), if you sample faster you are wasting bandwidth.
However, note that sampling is just the first step. You do not get a digital signal after sampling. You actually get a discrete-time signal. To make it digital, you need to quantize the analogue value you have recorded for each sample. In other words, you have to map each value to a combination of bits. This two process usually happen together tho... your ADC circuitry will allow you to specify the sample rate and the number of bits to use (that may be fixed in some cases).
So, for instance your sound card have an ADC. If you record an audio file at 16KHz 16bits, you are sampling at 16KHz (so you are capturing signals up to 8KHz) and quantizing the samples using 16bits.
When you signal is already digital, as it happen with a computer generated value, you do not need to sample it, and the Nyquist theorem does not apply. The value you have is what you have...