Do I pay VAT on real estate agent commission and property management fees?

Yes. Brokerage commission, property management and service charges are taxable services at 5% VAT when the provider is VAT-registered (registration is mandatory once turnover passes AED 375,000). So even on an otherwise VAT-exempt residential deal, the agent's ~2% commission carries 5% VAT on top. Budget for it, and check invoices show a valid TRN. General information, not tax advice.