Understanding singular vs plural: How to use words and phrases 'group', 'number of' and more

Should ‘group’ as a subject in a sentence be used as a singular or plural? Is the sentence “The group were very happy to meet after a gap of 10 years” grammatically correct?

‘Group’ can be treated as a singular or as a plural depending on the context. If it is considered as a whole or a unit, it can be treated as a single entity. Here are examples:
1. The group has decided to postpone the training programme to 21st February.
2. The group wants to meet next week.
3. A group of teachers has decided to donate their one month's salary for the cause of educating some poor children in their locality.
In the sentences above, the group is considered as one unit and therefore it is treated as a singular subject.

If the individuality of the members of the group is the focus, then it is treated as a plural. Consider these examples:
1. The group are in disagreement with the director’s proposal.
2. That’s a load of rubbish. I don’t think the group have any pretensions to be pop stars.
3. And the group have now celebrated their diploma success.
The first sentence implies that some members of the group agree and others disagree with the director. The third sentence implies that each member of the group celebrated their diploma success.

Collective nouns such as audience, class, crowd, family, committee, team, panel, and staff can take either singular verbs or plural verbs depending on the context. In American English, most collective nouns take singular verbs but in British English, collective nouns take plural verbs. Here are examples:
1. The Delhi government is interested in improving the quality of education.
2. The government have decided to introduce the bill in the Parliament.

The rule is that if we refer to the whole group as a single entity, then the singular verb should be used and if the individuals in a group are emphasised, then the plural verb should be used. Let us look at these examples to understand this concept better:
1. The family is very much united. It eats and prays together. (Singular verb)
2. The family quarrel among themselves frequently. (Plural verb)

Is the collective noun phrase ‘number of’ followed by a singular verb or a plural verb? If the ‘number’ is preceded by ‘a’ in a sentence, then it takes a plural verb as demonstrated in the example below:
1. There are a number of research papers on the topic.

If the ‘number’ is preceded by ‘the’ in a sentence, then it takes a plural verb as demonstrated in the 
1. The number of students attending the course has come down.