Case of sentence formation. If the precedant is in the past perfect tense, the antecedant must be in present perfect tense. Hence, option 2.

Case of coordinative conjunction. 'Lest' is always followed by 'should'.

Serving our parents is a 'responsibility' that comes from within. Modal 'should' is the best choice to depict a responsibility.

This is a tricky sentence. In sentences like this, we prefer usage of 'dare', but this sentence is followed by a question mark (?) and thus, 'can' is the preferred usage.

This sentence demonstrates the 'ability of past'. Modal 'could' (which is past of 'can') is the best option for the same purpose.

This is an imperative sentence. Somebody is given an advice for not to do something. Moreover, the sentence is in the present tense. So, modal 'should' is the best option.

'Hope' means probability. And for probability we use either 'may' or 'might'. Verb 'thought' in the sentence shows that the sentence is in the past and 'might' is the past of 'may'.

This is a case of 'seeking permission', and no other option than 'may' fits to the requirements of the blank. But when it is a case of 'speaking', we prefer 'can'.

A piece of advice. It shows that it is our responsibility to pay our debts. Thus, option 2.

This is a case of 'hope'. In this case there is a hope that landing of the plane can happen at any time. For conditions like this, we prefer 'may'.

Case of seeking formal permission. In such cases, we use 'may'.

In cases of ethics or morality, 'ought to' is a preferable choice. Moreover, helping the poor is a sort of responsibility (social responsibility). Modal 'should' is the best options from the given ones.

'May' is used in case of seeking or giving permissions.

Whether one keeps one's words or not depends on many factors, of which 'personal will' is most crucial. When we say 'rest assured',  we mean 'if everything else taken care of'. In such cases, keeping our words is just a function of our 'willingness' and when we are sure, we use modal 'will'.

This is a case of 'advice'. Whenever we advise somebody, we use modal 'should'.

Occurence of rain is always a matter of probability/possibility, and never a matter of certainity. For showing probability, we use either 'may' or 'might'. In this case, the information that 'the sky is overcast' add to the probability. Thus, 'may' is preferred to 'might'.

The sentence implies that for the crops to survive, rain is urgently required. This is a sort of compulsion. Modal 'must' best fits to the blank.

It is a case of advice. 'Need not' is the best usage in such cases. It means 'you are not supposed/required to'.

Solving a problem depends on one's capability. For capabilities, we use either 'can'  (present tense) or 'could' (past tense). Verb 'tried' means past tense, so 'could' is the preferred usage.

To pass an exam is a matter of probability, and for probability we use either 'may' or 'might'. But conjunction 'so that' is followed by the first form of the verb. Hence, 'may' is the correct usage.

The sentence is in the past tense ('did'). For past ability, we use 'could'.

Usage of exclamatory mark at the end shows that this is a case of 'a wish'. In cases like this, modal 'may' is used.

This is a case of 'seeking permission'. We use either 'may' or 'could' for the same purpose.

In cases like this, no mood is expressed and it is a case of probability of happening of an action. In such cases, we use 'will' as an auxiliary, which means 'nobody is sure'.

The word 'lest' is always followed by modal 'should'.