We use why is short replies to affirmative statements. We use why not in replies to negative statements.

Compare:

‘He has decided to quit his job.’ ‘Why?’ (NOT Why not?)
‘I can’t do this.’ ‘Why not?’ (More natural than ‘Why?’)

Why not can also be used to agree to a suggestion.

‘Let’s go for a walk.’ ‘Yes, why not?’

Why not + infinitive

Why can be followed by an infinitive without to. This structure is used to suggest that an action is unnecessary.

Why argue with him? He will never change his mind. (NOT Why arguing with him?) (NOT why to argue with him?)

Why not + infinitive is used to suggest something for other people’s consideration and acceptance.

Why not dismiss him from service?

The above sentence means ‘Why should we not dismiss him from service?’
This is a more pointed way of making a suggestion than ‘What about dismissing him from service?’

Notes
It is wrong to say ‘Why not we dismiss him from service?’. This is a common error which should be carefully avoided. No noun or pronoun should come after why not.

More examples are given below:

Why not buy and pay later?
Why not try it again?
Why not arrange a party in his honor?

‘Why don’t’ can be used in the same way. Note that it is followed by a noun / pronoun.

Why don’t you try it again? (NOT Why don’t try it again?)
Why don’t we go and see him?

Why should

The structure ‘why should…?’ suggests surprise. In American English, ‘Why would…?’ is used.

I wonder why she should get angry with me.

This structure can also suggest refusal to do something.

I don’t understand why we should have to pay for your mistakes.