Traditionally the UN covers about 75% of private school up to a limit.  Starting 2017/18 a new standard will be applied, in which different percentages will be applied for different tuition amounts.  Basically, the lower the price of the school the more coverage.  You can request advance for next school year and/or reimbursement from the prior year school year.

Funds that the UN employee receives as help to relocate to New York.  Keep in mind that it may take weeks for the UN to disburse the funds, so you may have to advance your own money in the meantime.

The rental subsidy is a very important benefit for UN families and you should plan on using it. Note that the rental subsidy amount is not fixed, it depends on how much the rent is.  The subsidy increases as the rent increases, up to a maximum amount.  The subsidy also increases as family members increase (except for the first child, where the subsidy is the same as with spouse only) and it also depends on you grade level.  For example, if based on your grade and family size your maximum reasonable rent level is $5,500, you can rent for $4,000 and get $1,000 in rental subsidy.  Or you can rent for $5,000, and get a $1,800 subsidy.  So the difference for you between renting for $4,000 or $5,000 is only $200 from your own pocket, not $1,000.  Make sure you play with the numbers to see what rent level is best for you, and take into account that the rent may increase 3%-5% a year.  

The UN will help you pay for part of the broker fee one time, check on the website above to see what the broker fee coverage is.  Basically the UN covers 80% of the broker fee above you "threshold amount" up to a limit.   This is important because broker fees vary in NYC; you can always ask your broker what their fee is.