she is very friendly and gets on well with her classmates中的gets为什么不能是getting而she likes eatting and cooking中的cooking又用了ing形式

可以拆成，she is very friendly and she gets on well with her classmates。这句的主语是she。

and 连接的系动词和动词主语都是同一个呀，she
like doing 是固定搭配，喜欢吃和喜欢做饭

and 后面是并列结构，后半句相当于she gets on well with her classmates前面用一般现在时，后面也用一般现在时。cooking是动名词，在句中做宾语，所以是动名词的形式，这两个句型是不同的。

因为get的主语还是she啊 and前后为并列成分 she后是一个be动词 然后and连接一个实意动词 也就是句子可以拆为：she is very friendly. She gets on well…
明白么
She likes eatting and cooking中 eatting 和 cooking 都是like 的宾语，这里的and 连接的是两个宾语

在第一个句子中get 是作谓语,与is 一起作谓语,所以动词要用第三人称单数形式.
而在第二个句子中,因为已经有一个谓语动词likes了,所以后面只能用非谓语动词了,所以要用动名词作宾语.

前后是两句并列的短句。
she is very friendly
she gets on well with her classmates