Does the deviation from the center frequency depend on the frequency of the audio/information signal applied to the carrier?
No. In FM terminology, the "deviation" is chosen by the designer (or mandated by regulation) and determines the amount by which the carrier itself is modulated at the "peaks" of the input signal.
For example, if the deviation is 2.5 kHz (NFM) then at the moment when a line level input is +1.736V, the output frequency will be 2.5 kHz more than the center frequency. At 0V the carrier will be at that instant at the center frequency. And at -1.736V the carrier will be deviated from its center by -2.5 kHz. (For broadcast FM the deviation is usually closer to 75 kHz.)
So it might seem that the input signal has no effect on the bandwidth, just the deviation, then? But that can't be!
If that were the case then we could use, say, a 1 kHz deviation to encode a lovely audio signal up to 20 kHz. And if we could do that, why couldn't we keep extending that even further and use, say, a 100 Hz deviation to encode a 1 Gbps internet signal?! There's a reason that FM broadcast uses a channel spacing of 200 kHz, i.e. more than double the 75 kHz deviation.
On average whats the width of the FM sidebands?
Because of the maths of having a signal that's essentially a cosine of another sinusoidal function, it's a bit complicated. And there's no real stopping point either! Imagine something like a bell curve — or in this case, a Bessel function — that gets gradually less significant away from its peak but never actually goes to precisely zero.
So FM broadcast uses an approximation known as the Carson Bandwidth Rule which says that the bandwidth needed is roughly $2 (\Delta f + f_m)$ where $\Delta f$ is the deviation and $f_m$ is the highest frequency of the input signal. Each sideband would be half of this overall bandwidth.
So that's the catch! The deviation itself isn't affected by the signals transmitted but the bandwidth used ends up being roughly the deviation plus the frequency sent (and then double that combined sum, since there's both an upper and a lower sideband). Using only 1 kHz FM deviation to transmit a 20 kHz audio signal still results in a bandwidth of about 42 kHz ignoring ~2% of the signal power (which ends up even farther from the center frequency).
Finally, note that an AM signal has sidebands too! Even though think only of the carrier amplitude changing and not its frequency, the change (modulation) itself adds information to the signal and necessarily increases its bandwidth. This is counter-intuitive at first, so when thinking about that it might help to remember the above. In FM the bandwidth is wider than just the deviation, too!