1) You could have ripple (power line hum) in the power you are feeding the TDA2005's. 1000uF is a very low value for a power amplifier filter capacitor, one would expect something 10x to 50x larger than that. Try putting a MUCH larger capacitance there.2) You could be picking up hum due to inadequate shielding of the audio input (which is rather sensitive.) Try shorting the power amp audio input directly to ground right at the TDA2005 chip. That would be one good differential diagnosis for seeing whether the hum was internal (from the amp power supply) vs. external (picked up in the audio input wiring.)
Yes, but take that almost non-existengt ripple and multiply by your audio gain of 50dB per the data sheet, on the order of 10-20,000x, and now that ripple becomes pretty audible.