The fact is that cable quality and noise DO indeed affect digital signal quality, and it is not 1s and 0s being transferred along the cable. Here's why.
A cable has capacitance, and that capacitance turns a nice, neat binary 'square wave' signal...
...into something called a sawtooth waveform...
When the peak of the 'sawtooth' doesn't quite reach the level required for the receiver to register a 1 or a 0...
...the transmitted signal now has an error because the signal voltage doesn't get high or low enough for long enough for the receiver to register it as a 1 or a 0...
As the capacitor of the cable charges up, its resistance increases, which is why the square wave gets turned into a sawtooth. The longer the cable, the greater the capacitance and the lower the data rate that the cable can support. Modern data cabling of the type you'll find in offices will transmit and receive at 100 megabits per second over a distance of around 100 metres.
However, to overcome this, digital signals are modulated. If you've heard a fax machine or an old dial-up MODEM working (MODulator/DEModulator, turns digital into analogue), those tones are modulated data. But it's not that simple. That kind of modulation works up to 9600 bits per second or thereabouts. So to get higher transfer speeds, the modulation is a bit fancier than just changing frequencies for different bits of information. High speed modems change the frequency (pitch) the amplitude (volume) and the phase (angle) of the signal in order to encode multiple bits in each state change.
USB v1 runs at 12,000,000 bits per second
USB v2 runs at 480,000,000 bits per second
USB v3 runs at 3,000,000,000 bits per second
Higher speeds demand higher quality cables and shorter cable lengths. The actual transmission rates are much lower than the speeds you think you're sending data at, because as I said, multiple bits of information are sent in each state change. This allows manufacturers to develop higher apparent data speeds from cables which haven't really changed much in the last 100 years. Copper is still copper.
Poor transmission media will definitely cause errors. In the old days, this would mean that, for example, a large data transfer would fail part way through and have to be restarted from scratch. I sat on site until late into the night at a factory where a 6 hour print job would fail part way through. The cause was a single bit error, turning a hex '11' into a '16' so instead of the printer knowing to start a new line, it just carried on churning out complete crap. The cause of the bit error was noise on a piece of cable running through the factory.
Fortunately, modern transmission systems have error detection and correction mechanisms which work in two ways, firstly sending redundant data and secondly providing a feedback mechanism to retransmit errored data. These mechanisms will correct any transmission errors, however in doing so, they introduce delay. When you print a document, it doesn't really matter how long it takes the data to reach the printer, so delay is not a problem.
For real time audio, though, that delay means that the receiver has to wait to catch up with the retransmitted data, and if the receiver's buffer runs dry, the audio device will have no numbers to feed into its CODEC (CODer/DECoder, turns digital to analogue) to produce music. I'm sure you've heard a CD player that sounded like a stuck record, playing the same couple of seconds, over and over. The solution is to transmit data faster than the receiver can use it, so that the delays caused by retransmission are always absorbed by the buffer.
So a 99p cable? Definitely you will get errors and noise, and the longer the cable, the worse it gets. A $1500 cable? Yes, that is ridiculous. But if you're buying a $100000 audio component, you want to feel like you have spent a suitable amount of money on accessories, so I'm sure it looks and feels just great. If you're the kind of person who goes into Harrods and says, "I'll take it" then this is a cable meant just for you. You have no idea what you're buying, so if it's expensive it must be good. You'll never listen to it anyway.
However, in terms of the above explanation of data transmission, a $10/£10 cable would be more than sufficient.
Analogue cables are something very different. The chemical composition of the cable affects its resistance, capacitance and inductance, and these characteristics change at different signal frequencies. What leaves your source device is not what arrives, one metre later, at the amplifier. The frequency range is 'smeared' and different frequencies are amplified (louder) or attenuated (quieter). Try singing in a tiled hallway, bathroom or sports hall and notice how your voice sounds different - that's what happens along the length of the cable. However that doesn't make the sound necessarily worse, it makes it different, and so analogue cables have to be matched just like other components, to create the sound that you like.
Ultimately, the original source of music is generally analogue (voice), and the reproduction is always analogue (ears) so the analogue connections will have a much greater impact on subjective sound quality than the digital ones.