I think its possible with a PSoC, but not directly with the out-of-the-box components. There is a filter component which does what you need, but it can be configured onlty during build time. There was a discussion here http://www.cypress.com/?app=forum&id=2232&rID=56121 where you might follow up on how to configure the filter during runtime.
What might work is to implement a digital filter on your own, but this might get complex...
A suggestion would be to ask here: http://www.cypress.com/?id=2820 .
And to answer at least one question positively: yes, there is a quadrature encoder readily available, together wird modules to work with text or graphical (TFT-like) displays.
Is it possible to build a tuneable filter with a PSOC ?
Yes it is possible to build a tuneable digital filter with PSoC3.
How many different filters can you run simultanously on a PSOC device ?
The digital filter block in PSoC has 2 filter channels that can run parallely.
There is a quardarture decoder in PSoC Creator library. I am afraid there is no Quadrature encoder at this point of time.
You can take advantage of the power of DMA in PSoC to switch between different filters(Lowpass, highpass, bandpass, bandrejection etc). The digital filter blocks makes use of a specific set of coefficients in its own RAM to perform filtering. If you can load the RAM with suitbale coeifficients you can change the filter characteristic in runtime. DMA comes handy in this aspect. The gain can be adjusted by adjusting the coefficients. 44.1KHz audio seems possible. But getting a analog output with a 12 bit resolution might not be possible with the exisitng DACs in PSoC. You need to pla around a little bit to achiev 12 bit DAC from 8 bit DAC, then again there are different challenges associated with that.
Hope this helps...
PSoC3 or PSoC5 should be your choice for this application.