Can sound generated by interference of cell phone signals contain any useful data?

Short answer: No.

Long answer: Your accidental receiver can't see the data that's being received. Also, you're listening to an effect of the uplink, not the downlink. Also, cryptography.

Longer answer:

What you're referring to is interference where you can hear the power modulation by data bursts emitted by your phone.

Why your phone and not the mast? Your phone is way, way closer to you, and power density drops with the inverse of the surface of a sphere with the distance as radius. And that's quadratic.

The base station your phone communicates with will tell the phone pretty early on how to regulate its power so that the signal is strong enough, but not too strong, when it reaches the mast. The other way around, it does the same: a GSM tower will not send vast amounts of RF power for your phone; it will send at most as much as necessary to your phone that at the point that it is, it has good reception.

The powers for "good reception" are due to the "power goes down with the square of distance" argument above very much lower than what the other side has to emit. Therefore, your radio receiver doesn't even remotely notice the far-away transmitter. It only "sees" the phone right next to it. And it really needs that close-by transmitter to be strong (see next paragraph).

Why power modulation? Because you're not actually using a receiver for GSM or even GSM frequencies, your radio receiver is just not sufficiently protected against interference from other frequencies¹. You don't have a demodulator for the data, it's just the nonlinearities in some component that mix down the signal to baseband.

Thus, what you hear is the rapid succession of "data burst" – "silence" – "data burst"... ; the symbols in the data burst itself are too short to be audible². So, the fact that you hear something is not the actual data in GSM, it's that a burst is 577 µs long (and consist of 156 symbols or so), and followed by silence. That's what makes the sound while you're having a call!

Now, the data modulation itself for GSM is GFSK³, which is essentially a constant envelope modulation, which means that after (inadvertent) mixing to baseband, even if you had a 280 kHz observation bandwidth, all the data would have been mathematically erased – there's no mutual information between the data in the symbols and the baseband result.

You can build a proper RF receiver for GSM for 10 to 30€⁴. Then, your computer would get a mixed-down version of the signal that retains the phase information, and thus, can actually be used to recover the bits that were sent. There's software for that!

The number of the caller belongs to the things that aren't necessary to communicate at any point to establish the connection, so they're not sent at all: A phone number is not really a concept to a GSM network - it doesn't care about phone numbers, really, until they are necessary to know where to send call data, and that's very far away from the RF interface.
As a subscriber, you're identified by other means. So, your phone doesn't send its phone number at any point, far as I know. There's no reason it should.

The phone number of the caller is optional, and only received by your phone; but you don't hear what is received, for power reasons as explained above, you only hear when something has been transmitted by your phone!

However, as said above, GSM networks are encrypted. Even if you have a proper receiver for GSM frequencies, properly lock on to the network, properly demodulate GSM messages: you still have to break the encryption on these. That's not impossible for a couple of the older encryptions that exist in the GSM standards, but it's still significant effort that requires a bit more understanding.

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¹ as it should be – modern high-fi stereos with high audio gain probably are well-designed to keep typical domestic sources of RF power out of their signal path, say phones, wifi, bluetooth, 700 / 900 MHz ISM band automation...

² that would require audio to go up to above 280.333 kHz, the GSM symbol rate.

³ I think 8PSK might also be standardized, but that would be another constant-envelope modulation

⁴ 6€ if you directly order an RTL-SDR dongle from China.