Originally, cool meant someone who was somewhat standoffish in their demeanour, or someone who does not get too involved - and cold was just a more extreme form of that.
As Bill Franke points out, the meaning of cool meaning popular or trendy probably originated with cool jazz in 1945 and began being used to describe people in the 1950s seeing a rise in usage through the 1960s and a large spike in popular usage in the 1980s and 90s:
For this reason, "he's a cool guy" in modern English means "He is a trendy (or fashionable) person" rather than "His character is standoffish", whereas the meaning for cold never evolved to mean trendy, and retains its older meaning of someone who is very detached and unemotional.
If you want to use cool in the sense of someone who is slightly unemotional, you might prefer to use the idiom He's a cool cat or He's a cool character which both retain the old (non-trendy) meaning of the word cool. Alternatively you can say something like His response was somewhat cold as a less ambiguous alternative to His response was cool.
What great examples!
As for the last set of sentences: I'd call them just about interchangeable, except that "a lot" would be regarded as less formal. (NOAD tags "a lot" as "informal," e.g.) If I was proofreading my own writing, I'd probably change "I don't know a lot about" to "I don't know much about." Either would be fine in conversation, though.
I don't know much about Western history. {okay}
I don't know a lot about Western history. {okay in conversation, perhaps a bit informal for writing}
As for the middle set: I'd rephrase the latter sentence to read, "I don't watch a lot of TV." The first one could also be rephrased ("I don't watch very much TV"), but those two sentences have slightly different meanings: "I don't watch TV very much" would probably be interpreted as "I don't watch TV very often"; while "I don't watch very much TV" sounds more like "I don't watch many TV programs". The difference is very subtle, but I think it's existent. If I watched a half hour of TV nightly, but always the same program, I'd be inclined to say, "I don't watch a lot of TV," but not, "I don't watch TV very much." (After all, I watch nightly!)
I don't watch TV very much. {you don't watch TV very often}
I don't watch a lot of TV. {you don't watch too many TV programs}
I don't watch TV a lot. {not wrong per se, but I'd recommed one of the others}
As for the first set, that latter sentence sounds off to me. Interestingly enough, I have no problem with its inverse: "I like Japanese food a lot." However, when speaking in the negative, "I don't like Japanese food very much" sounds much more polished than "I don't like .. a lot."
I don't like Japanese food very much.
I don't like Japanese food a lot. {use the first one, not this one}
I like Japanese food very much.
I like Japanese food a lot. {in the positive, you can use either one}
I'm marveling how, even though the three pairs of examples all have the same sentence structure, my comments are different for all three. I guess this is a trickier problem than one might first expect.
Best Answer
In short: you're correct.
Long-form: The use of "so" in
is extremely idiomatic, and informal. It's used to emphasize the degree to which "you" are "not cool". The result is a superlative statement, in a way, declaring the level of "not-cool-ness" to be extreme, and beyond measure. Verbally, the word "so" would be strongly stressed.
Compare:
This construct would be well placed in the film "Mean Girls", delivered as an insult with a dismissive and exasperated affect.
Conversely, placing "so" after "not" is much more thoughtful, (though still critical.) In this case the sentence suggests that the listener has an overly-inflated ego, or perhaps that the speaker formerly thought that the listener was once a "cooler" person than they are now perceived to be. The "not" is used as a polarity switch, and "so" is used as a modifier of "cool".
Compare:
This phrasing would likely show up in a light comedy scene, as a form of criticism, but lacking malice. It could be said between friends, and be at least forgivable, or maybe delivered and taken entirely as a joke, or "ribbing".
One may notice here that, as is common in English, both examples are used as shorthand to express a longer idea, which I spelled out in [square brackets], making them both "jargon", and incomplete in themselves.
Citing my source: I'm a native speaker of American English, raised in California, with some college, several years of customer service communication, and an author for a parent. Practical linguistics is a hobby.