The correct phrasal usage is 'dare not.

Whenever we refer to a work of future in reference to the first person, we prefer 'shall' to 'will'.

To ask somebody's choice on a mattter, we use modals 'will' and 'would'.

 We always keep moral duties above a responsibility or a duty/an obligation, so usage of 'ought to' is correct.

This is a case of present indefinite tense. As the sentence shows a sort of certainity for the action to be done or not to be done by a third person, so 'will' is the correct usage.

For asking somebody's choice on an issue, we use modal 'will' or 'would'.

'Obeying the parents' is not a duty or an obligation, rather it is an ethical responsibility that comes from within. So, 'ought to' is the correct usage.

As the sentence says that the old pen was still working fine, so it simply clearifies two things - 1. The sentence is in the past tense, 2. There is no need to buy a new pen. For past, we use 'did'. Hence, option 1.

This is a case of past tense. None of the given modals fit the blank. Here we need auxilary, i.e. option 4.

The sentence refers to 'likely happening of an event'. 'Should' expresses the same.

The sentence implies that an activity or fact of the past no longer exists. And for such cases we use 'used to', which means that such things prevailed once upon a time, but not in existence now.

The sentence refers to a 'scheduled event', i.e. the statement making. In such cases we use 'is to'.

This is an assertive sentence where the probability of aphenomenon is asserted. In such cases we use modal 'could'.

Being punctual is a sort of responsibility, more than being an obligation. So, modal 'should' is the correct usage.

Whenever the first person seeks a permission, usage of modal 'shall' is preferred.

In the sentence usage of preposition 'to' just after the blank eliminates the chances of all other options than option 1 as 'used to' is the correct usage.

The sentence implies that 'he' didn't have enough courage or audacity to ask for a raise in salary. In such cases we use 'dare'.  

For referring to wishes we use modal 'would', and auxilary 'were' (if required).

Whenever we ask for somebody's opinion on an issue, we use modal 'will' or 'would'.

HEre we are referring to an ability of the past. In such cases, we use 'could'.

Case of perfect tense. In such cases modals are followed by auxilary 'have'. Thus, option 4.

This is an indirect speech. Whenever reported speech is in the past, reported verb is also in the past tense. 'Would' is past of 'will'.

The sentence refers to the probability of a phenomenon. And for 'slight probability', we use modal 'might'.

The sentence refers to a habit/fact of the past, which is no longer in existence. In such cases, we use modal 'used to'.

Whenever we refer to a specific time of the past, we follow rules of past tense. 'Am' is used for present tense, so option 1 and 3 eliminated. 'Was to have left' means 'upposed to have left'. Thus, this is the right choice.