Both amplitude and frequency modulated radio signals nowadays use quadrature modulation and demodulation as a mean to transfer and receive radio signals.
This question is ill-defined – a signal doesn't use quadrature modulation/demodulation, transmitters/receivers do.
So, your question is, if we try to "rescue" it, is
Do modern FM receivers/transmitters use quadrature modulators
No, not generally. When I receive FM audio with my USRP, I might do that, but usually, I will use a trick and actually do a low-IF sampling.
When a car radio receives signals, it definitely doesn't – simply because IF receivers typically still have technological advantages over direct conversion /IQ receivers, mainly due to lack of DC offset, and because you can get high-Q analog bandpass filters easily. The bandwidths used for FM audio are so laughably small that any modern 16 or 32 bit microcontroller can usually deal with the IF sampling rate.
Now, assuming this is about FM audio broadcasts only: I'd suspect that you'll actually see the different components of the signal first being generated digitally, then digitally mixed to different IFs, and then generated using IQ DACs and mixed up using a Quadrature modulator. But: Broadcasting equipment often has seen quite a few winters, so this might not always be true!
Whenever quadrature modulation is used the sampling rate should be equal to bandwidth.
No, not generally. Nyquist says that in the complex sampling case, it is sufficient to sample with nothing more than the signal bandwidth, but in fact, you rarely do that, because you then wouldn't have the chance to suppress noise due to oversampling, or to have "spare" bandwidth to do frequency correction. Also, there might be simple technical problems ("can't get an oscillator running at 13.13131313 MHz") that make it impossible to sample at exactly the bandwidth. And, there's also applications where undersampling is employed.