Both 愿意and肯are operative verbs in Chinese language which are used before the verbs. But there are some tiny differences between the meanings and they can’t replace each other. The former is applied to express people’s psychological feeling, which is to say, somebody is willing to accept and even love to do something, without unhappiness or being forced. However, teachers in Chinese language school may tell you that 肯is to express somebody accepts something through language and action. Such kind of reception needs to be achieved by making certain efforts and even some sacrifice.
When we use the word 肯, there are usually two conditions, one of which is to do something without wiliness or under the demands of others. The second condition is to do willingly without being demanded. But it usually means the actions or behaviors in usual time. In addition, the adverb of degree like 很and非常can be used before 愿意to modify it while 肯is seldom modified by the adverb of degree. For example, 我愿意娶张舒雅为妻and他很聪明有肯吃苦，所以很快就掌握了基本功.
Let’s move our attention to the differences between 应该and必须. Since both words are different in semantics, they can be replaced by each other in sentences. The meanings of the sentences 你应该去and你必须去are different. When you study in intensive Mandarin course in China, you may learn that 应该is to indicate that the speaker thinks it is correct to do in certain way, explaining it is the suggestion or opinion about something, equivalent to “should” in English.
However, 必须indicate that you must do in certain way and there is no other way to do certain thing, which is the same as “must” in English. 必须are usually applied in two situations, one of which is to give order and the other is to describe objectively that someone can do only in certain way is certain conditions. When it comes to this situation, the overseas students often mix it with 应该. In addition, 应该can be applied to indicate the speaker makes a guess to certain condition but 必须has no this application.