Does the 'and' have to be replaced by 'or', and if so, can you explain why?
"or" certainly parses better, since "don't like" refers to each individually, rather than presenting an "and" list. IF I were editing, I would certainly change the "and" to "or", but that's just me.
"Brahms, Mozart and Dvorak" are composers I don't like." is a list, so "and" is correct.
I wouldn't say it's optional.
"I speak English and Chinese well. I don't speak Russian or German."
If I said, "I don't speak Russian and German," it would suggest I don't really speak English well.
I think the 'or' is compulsory after a negative - nothing else would sound right.
You can say "I dislike B, M and D" but if you say "I do not like ...." there has to be an 'or'.
Edited by: libbyh
Yes, I agree with #3. Or is usually compulsory after any negated verb.
The only (rare) exception would be when you are talking about two things that can construed as a single unit: e.g. "I don't like John and Mary" (meaning the obnoxious married couple who lives next door) or "I don't like fish and chips" (meaning the combination served as a dish).
99% of the time, though, or would be the norm.

Though you can say: I don't like Pinky and Perky. Presumably as Pinky and Perky are seen as one 'unit' rather than two separate entities. But then how do you decide who or what is a 'unit'? I don't like Mick Jagger and Keith Richards as they are both very naughty boys. Acceptable?
What's acceptable? There's no firm line. It depends on the emphasis.
"I don't like John or Mary" stresses you don't like them individually. "I don't like John and Mary" stresses your dislike of them as a couple.
If you say "I don't like Mick Jagger and Keith Richards," you're suggesting it's the two of them together you don't like.