As a general rule, no. The exception is where there is an agreement by both parties to the employment agreement. This is best done in writing to avoid any later dispute.
Yes, provided the employment commenced on or after 1 October 2000 (the date the Employment Relations Act 2000 became law). If employment commenced before that date, then a written employment agreement is not mandatory, however an employee can request one and it then becomes mandatory.
Before it is signed, yes. Negotiation, however, does not necessarily mean that both parties have to agree. Nor does it mean that one party can impose the agreement unilaterally.
It depends on the agreement. If there is no mention of redundancy in the agreement, or the agreement specifically states that compensation is not paid, then compensation is not payable. However, where both parties agree, compensation may be paid.
It depends on how confident you feel about presenting your case and getting the result you are looking for. Representation tends to increase the likelihood of a successful outcome at mediation and at the Employment Relations Authority.