这里考察insist的用法：
insist 用法与suggest 相似.
insist作“坚持”说时,要用陈述语气；
作“坚决主张；坚决要求”讲时,用虚拟语气.如：
They insisted that he （should）be present at the ceremony.
他们（坚决要求）他出席这次典礼.
He insisted that he was innocent.=He insisted on his innocence.
他（坚持说）自己是无辜的.
第一题：1.Mike's father,as well as his mother ,insisted that he ___home.
A.Stayed B.could stay C.has stay D.stay
意思是迈克的爸爸妈妈（坚持要求）他留在家里.表示“坚持要求”的时候要用虚拟语气,should+do,should常常省略,故填动词原形,故选D
第二题：
2.The boss insisted that Mary____fit for the job because she was thought to be unable to bring in money for the company and he suggested that she ____ at once.
老板（坚持说）【不是坚持要求的意思】老是说玛丽不适合干这份工作,因为她没有能力给公司带来效益,所以老板【建议】立刻辞退她.
suggest的用法：
1.suggest 作“建议”解,而且后面跟 that 从句作宾语时,宾语从句中的谓语动词用虚拟语气,即谓语动词用“（ should ）动词原形”.
2.suggest 作“表明、说明、暗示、指出”解,而且后面跟 that 从句作宾语时,宾语从句中的谓语动词用陈述语气.例如：
① He often came to the theater, but his attitude suggested he was not really interested in the play.
他经常去剧院,但是他的态度【表明】他对那个地方不是很感兴趣.
② Jane's pale face suggested that she was ill, and her parents suggested that she have a medical examination.
简苍白的脸【表明】他病了,他的父母【建议】（表建议时用虚拟）她去做个体检.