In addition to income tax there is a government ACC (Accident Compensation Corporation) levy that is deducted from your salary by your employer. You basically pay 2% of your salary. This covers you for treating injuries and issues caused by accidents (as the name suggests). More can be found out about it
here. Note it is a
government levy for public healthcare. It is not driven by your employer for private cover.
In addition you may purchase private health insurance but most don't.
In terms of public holidays, there are
up to 11 days a year, but if some public holidays fall on a weekend, they are not observed (or "Mondayised").
Dept of Labour lists them.
The answer to question 4 may not be in your job offer as some benefits (and it depends what types of benefits you're talking about) are not contractual. Those that are contractual should be in with your job offer. But if you're looking at benefits like discounted health insurance, cheap cars (yes in my old company we got discounts off Vauxhall and Saab cars!), flexible working arrangements, etc, these will be available in company policies that will be available when you start work. Unless you make a specific request to the HR team at your prospective employer.
In my experience, those nice-to-have benefits are less common here than you might find in other countries. But it completely depends on the company you're working for. If you're going into the banking or financial services sector, you will likely get preferential rates on those types of products. If you are going to work for Air NZ, you'll get staff travel after 6 months. I'm sure if you work for a telecom company, you'll get cheap deals on their products. Government department employees also get some good benefits like subsidised school holiday programmes and cheap creche facilities. But there really isn't a "standard" of benefits.