does "和" between verbs change them into nouns?
As I know "和" (he2) stands between two nouns and means "and". But I found in my chinese book the phrase: "听和说" what I think means "listening and speaking", but "听" is only a verb "listen" and "说" is only a verb "say". So my question is: is there any grammatical rule for this?

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listening and speaking，强调过程，但也是动作，基本属于一个持续性动作。
listen and say ,强调动作，不强调过程。
听和说，强调是过程，同时也是动作，所以你理解它们是"listening and speaking"是对的。
听和说，分别是listen and say ,也是没错的。

不知道我用中文回答，你能看懂不能，如果不能，可以给我发信。messenge to me.

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or you can try to understand it in this way:
as you have known, “听和说”means "listening and speaking", does listening still a verb? or can listening be used as a verb? listen is a verb, but with -ing, listening is no longer a verb in my mind, because it can not be used as verb in sentence. can you catch what i mean?

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No. 和can not change verbs into nouns.  
和 can be  used between two nouns or two verbs or two prons, but it doesn't mean the two verbs with 和 in the middle become nouns. 
two verbs: 吃和睡
two nouns: 书和本
two prons: 我和你

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an Chinese verb usually has a corresponding noun to it under given condition! 
and sometimes it also can be seen as either verb or noun.

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you can regard them as nouns or verbs that is not important.  听和说 also can be considered as listen and say. chinese emphasize meanings but not forms. 说、学、逗、唱 can be nouns too. you can even use adjectives as predicate. e.g. 我的脸啊红啦.  if you use english grammar to explain that, 红 here is a verb meaning turn red.