Is it true that a cassette recorded at a faster speed will have a
wider frequency response, richer S/N ratio, and an overall better
quality than a cassette recorded at a slower speed?

If the machine is designed to have bias and eq settings appropriate to
the higher speed, yes. Some of the 4-track "Portastudios" used
cassette running at 3 1/2 ips and a non-standard 4-track format.