1.I think that Joe is no more like(ly) to win the competition than Zoe.
like是副词,likely是形容词
2.Look(ing) up new words will help you get the main idea of the whole passage very soon
look 和 will都是谓语,一个句子只可以有一个谓语,Look(ing) up new words就是主语了
3.The course on the study of collecting coins and stamps attract(s) a lot of students.
on the study of collecting coins and stamps修饰course,course是单数
4.Which group you will agree（with） is up to you.
agree with固定用法
我比较确定