1.在“Our English teacher does what she can (to help) us improve our English ''中what she can 省略了do 这个动词,what引导的是一个宾语从句,而 (to help) us improve our English 为目的状语从句.这个句子写完整就应该为：Our English teacher does what she can do (to help) us improve our English.
2.动词后什么时候跟to ,什么时候不跟to,需要平时的积累,如stop doing sth不做某事. ; stop to do sth.停下来做某事；remember doing sth.记得曾做过的事；remember to do sth.记得要做某事；还有一些动词只能跟doing,如finish doing sth.practise doingsth.等等; 有些只能跟to do .如：want to do sth.; hope to do sth.等等.
以上内容希望对你有所帮助.平时还望你多积累知识.