Under normal circumstances, the noise floor should definitely NOT be "part of the sound". If I can hear the noise at normal listening levels, of normal program material, the amp is faulty or badly designed. o___O
Same with frequency response. If an amp colors the signal to an audible degree, I'd say it does not fall into the "competently designed" category. An amp has to sound like a wire. :<
However, especially on AVRs in multi-channel mode, one needs to take into account that you don't have a lot of watts per channel, as usually the power supply is the hard limit here. So driving 4 low ohm speakers with an AVR in a larger room? Definitely not recommended.
This is also where more expensive equipment comes in: power costs. You could go the Class D route and hook up amps to the pre-outs but then you are already in a higher price range on the AVR alone, because manufacturers product segment pretty hard and know pre-outs are popular among enthusiasts.
IMHO the most important factor when it comes to value conscious equipment is: Know what you actually need! There is no point in shelling out for a 1KW amp, when you only want to listen in a small apartment at moderate volume. It won't sound "better".
Know your speakers (impedance, potential phase angle issues) and how hard they are to drive, know your room (listening distance, preferred volume), then calculate the approximate power needed and buy accordingly. For most of us apartment dwellers, an AVR is perfectly sufficient on it's own, even if transient peaks may occasionally clip, since that is largely inaudible.