Nyquist-Shannon sampling theorem states that sufficient condition for sampling is sampling rate higher than twice the highest frequency.
By highest frequency, does this refer to the highest frequency content of the signal's Fourier transform?
Yes. If the signal contains higher frequency parts, aliasing will occur and the signal will be distorted, this is why all digital recording equipment has filters that remove the signal parts above f/2. It is also why audio is recorded using at least 44kHz and not 40. If you recorded sound using 40kHz sampling rate and still wanted the frequency response to xtend to 20kHz you would need a filter with infinitely steep slope (pass 19.999kHz, completely block 20.000kHz).
It is also why recording using higher sampling rates is better - it allows the device to have a filter with less steep slope (for example, pass 20kHz, completely block 48kHz in case of 96kHz sampling rate) so there are less phase distortions.
If the signal has limited bandwidth and does not start at 0Hz (radio signals for example), then the sampling rate has to be greater than twice the bandwidth, but not maximum frequency (if the minimum frequency component is 1MHz and maximum is 1.1MHz then sampling rate has to be higher than 200kHz). For audio, the maximum frequency is the same as bandwidth, since audio usually starts at very close to 0Hz.
Yes, if the signal bandwidth starts at DC (0 Hz).
Otherwise, the highest frequency can be much higher as long as the bandwidth of the content is less than the sample rate, and the band doesn't cross N*(Fs/2) for any integer N. If there's no baseband signal, then the high frequency alias has nothing to alias against. That is how undersampling works.