I know you usually answer a question with 不 shi, 不 dui, 不 yong, 不 xing etc etc according to the situation and the verb used in the previous sentence, but is it possible to reply with only "不"? Or "不, wo xihuan......" (for example)? Or is it correct to say that that is gramatically wrong?

But does Chinese still say it? Or does this only occur as a 'slip of tongue', that is, not on purpose?
Is "不不不, ..." accepted in Chinese?

Of course. That's in all languages in the world.

Well, I hear it said alone all the time, or as 不不不。So it's definitely accepted in spoken Chinese.

take it easy ,many words in chinese can independently used even in written chinese so does verbal usage as well

it can be used any where except formal written chinese

yes

Does that mean that what they say: "Chinese don't have 'yes' and 'no'" is incorrect? Please explain in your answers.

不不不&不surely can be accepted,

but u'd better say:“不不不，谢谢。”（means:No,thanks).

that would more genteel.