First, "strength" is the power of the received signal as interpreted by the radio part of the receiver. This value is independent of "quality" but it affects "quality" directly.
Second, "quality" refers to how well the receiver is correctly decoding the digital part of the signal. Depending on the broadcaster up to 35% of the signal is redundant data called "forward error-correction." The quality is a numerical representation of both how well the FEC process is correcting for errors and how many errors are being encountered in the stream.
With these facts understood I have to say that lower quality signals can indeed be enhanced by an amplifier. It could be that errors are encountered, thus reducing the quality, because the signal strength is weak and is further attenuated by the cable length and switches, fittings, grounding blocks. The amplifier can help both strength and, indirectly, the quality.
As for interference and noise being amplified: I'm not sure what kinds of interference you would be experiencing in the 11 to 12 GHz range if your dish is aimed properly. There isn't much that can "interfere" except adjacent satellites and that is solved by adjusting your dish, the LNBF skew, or upgrading to a "low noise" LNBF. The amplifier, if introduced directly at the LNBF, would only be amplifying any noise generated by the LNBF itself as it downconverts to the 950-1450 MHz frequency range, and the only way to solve that kind of noise is by getting a better LNBF with a low noise figure.