1）源自于一个短语do well in,在...方面做的好,所以用do better in
2）一般在英语的文法中,宾语从句的引导词可以省略,i don't doubt that...这里的that引导的正是宾语从句；而后面的the rule is that...,that 引导了一个表语从句（在句子中充当的是一个表语的成分,因为the rule是主语、is是系动词,后面自然跟的就是表语了）,表语从句的引导词that是不能省略的.宾语从句和定语从句的引导词有时都可以省略.
3）prevent ...from...是一个固定搭配,表示阻止某人做某事.from可以省略.但在被动语态中,from无法省略,比如he is prevented from playing computer games.