Why do we quantize high frequencies coarsely, shouldn't it be the opposite? Also, in Huffman Coding, why do we use fewer bits for high frequency data. I don't get the intuition. Shouldn't it be the opposite or does this have to do something with the psychoacoustic model of the human hearing which says that our cochlea is sensitive to lower frequencies more than higher, hence we use up higher bits for lower frequencies?
In Huffman coding frequency refers to the number of times a particular sequence of discrete source symbols has appeared in the input. It is not the same concept as frequency in the continuous space such as sound frequency. As such, it has nothing to do with psychoacoustics or the cochlea. 

We use fewer bits for the higher frequency source symbols because frequent source symbols have lower entropy and assigning fewer bits to low entropy symbols gives better compression.