1. _____ man is a reasoning animal.
Reason: Man is a singular countable noun.
Question: As per article rule, "THE" is used when a singular countable nown is meant to represent "WHOLE" class.
Is using "THE" is wrong? Can I use "a" or "THE" ?
2. You should use ______ hammer to drive in those nails.
Reason: hammer is a single countable noun
3. ________ dead no longer need help.
Reason: I am not sure which rule of article apply for me to put "THE". I understand "dead" is an adjective and that the statement above is meant to represent "all" kind.
4. There were over ________ hundred people at the party.
Reason: Can you please explain me why? or why not "a"?
Here, man is representing the whole humankind. Therefore, 'the' is a more appropriate usage. However, using 'a' would not be wrong either. The reason being that the statement is trying to pose a universal truth about humans which can be done by treating 'man' as a singular entity as well as by treating it as a class.
2. You should use a hammer to drive in those nails.
The reason provided by you is partly correct. The article 'a' is used because this singular countable object is non-specific. If the sentence would have been referring to a particular/specific hammer then , 'the' would be used.
3. The dead no longer need help.
The reason is correct except that 'dead' here is used as a noun ( not adjective) to refer to the class of people who are dead.
4. There were over a hundred people at the party.
Here, 'a' is used to mean one i.e. one hundred people. Also, this can be seen as a phrasal use. For example: over a period of time, over a short span of time, etc. 'The' can not be used in any possible way.