In the simplest form, CW consists of a pure sine wave multiplied with a square wave that's either 0 or 1, corresponding to the keying of the carrier.
As with a mixer, or amplitude modulation, multiplying two signals generates frequency components that are the sum and difference of each frequency component of the multiplicands. In our simple form above, where the switching waveform is a square, the bandwidth can be very large, because a square wave consists of an infinite series of odd harmonics.
Real transmitters filter this square wave to some extent, and perfect square waves can't exist in practice anyway. The slower the transition from "on" to "off" is made, the less bandwidth is required.
If you send faster, then there are more transmissions per second. Since each transition requires a certain amount of energy away from the carrier frequency, faster speed means more sideband power, that is, more bandwidth.
Because every transmitter is different, it's difficult to say what the bandwidth is, exactly. We'd have to define more precisely what we mean by "bandwidth" anyhow. Because CW consists of brief periods of high bandwidth (the transitions) mixed with relatively long periods of zero bandwidth (everything but the transitions), the measured bandwidth depends greatly on how it's measured, anyhow. Are we looking at the average spectral density function over the entire transmission, or just some brief period around an on-off transition?
CW receive filters with a passband around 500 Hz are typical. It's certainly possible to make them narrower, but remember, it takes more bandwidth to make a sharp transition from "on" to "off". It doesn't matter if that bandwidth was never transmitted, or it was transmitted but we removed it with a filter. If the receive filter is too narrow, you won't hear a "dit dit" with clear starts and stops to the tone, you will hear a smeared "waahwaah". If we think about this problem in the time domain, it's called ringing, if we think about the filter as a resonant system, we say it has a high Q factor. This is actually a case of the uncertainty principle: the more sharply we localize a thing in frequency, the less sharply we can localize it in time.
I think the type, width, and nature of filters for CW is a matter of preference and circumstance. Between each pair of human ears is a wetware filter which is already very good at selecting tones by frequency, without help. Moreover, wetware filters can include context that simple linear filters can not: CW has a particular rhythm, QSOs follow a particular format, etc. However, a wetware filter can't remove a nearby strong signal causing desensitization.