A

we can use should after:

insist *「强调」 *demand「要求」 recommend「推荐」 suggest「建议」 propose「建议」

example:

• I insisted that he should apologize
• Doctors recommend that everyone should eat plenty of fruit
• What do you suggest we should do?
• Many people are demanding that something should be done about the problem

also:

It’s important/vital「必不可少的」/necessary/essential that … should…:

• It’s essential that everyone should be here on time.

B

You also leave out should in the sentences in section A.So you can say:

• It’s essential that everyone be here on time(=…that everyone should be here…)

• I insisted that he apologize.(=..that he should apologize.)

This form (be/apologize etc.) is called the subjunctive「虚拟语气」. It is the same as the infinitive「不定式」 (without to).「就像不定式在某些情况下没有to一样」

C

We do not use to..with suggest.You can say:

• What do you suggest we should do?

Or..

• What do you suggest we do?

But not

• What do you suggest us to do?

You can also use -ing after suggest(what do you suggest doing?)

D

You can use should after some adjectives, especially :

strange, odd, funny, typical, natural, interesting, surprised, surprising

• It’s strange that he should be late.He is uaually on time
• I was surprised that he should say such a thing

E

You can say ‘if something should happen…’

• We have no jobs at present, but if the situation should change, we will contact you.

  「我们当前没有工作，但是如果情况有变，我们会联系你」

You can also begin with should(Should something happen…)

• Should the situation change, we will contact you.
• 「如果情况发生变化，我们会联系你」

This means the same as ‘If the situation changes, …’ . With should, the speaker feels that the
possibility is smaller. 「意思是“如果情况改变了。。。”如果有should，说话的人感觉发生的可能性小一点」