In private equity funds, what costs and expenses are covered by the fund? Or, do the managers have to cover all the costs and expenses using what they get from management fees?

The fund managers (the GPs) typically cover the admin and overhead costs of running their shop. Traditionally, that's been the reason for the "2" piece of the classic "2 - 20" fee arrangement: to give the GPs a steady and predictable revenue stream from which to cover those costs. The carried interest component of the fee structure is a bit too volatile to serve that purpose.