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.
You're confusing the meanings of 'count'
Count with my fingers = enumerate… 1..2..3..
Count on me = depend on.
See OALD & check the difference in meaning between 1,2,3 & 4,5,6.
To say "It is from a friend you can always count with." would imply they would be quite happy to stand with you & help you decide how many oranges are in your shopping bag ;)
Best Answer
Looking at the Oxford English Dictionary, 'much' has many possible functions in a sentence. In the usage in the question, it would be a pronoun - substituting a noun phrase. One of its many definitions in the OED is C. 1a:
So to answer the question: yes, you can use it as you do in that sentence. However, both the OED and the experts from the Cambridge Dictionary state that it is used more in negative phrases.