I am an exempt employee of a privately-owned company. I occasionally go to trade shows during weekends. If I need to work during weekends (sat-sun), will I be able to ask my employee to allow me to take two days off during the weekday after my weekend job is done?

You can certainly ask. But it will be entirely up to the employer whether to agree or not, and it will be legal no matter which way they decide. Comp time is legal but not required for exempt employees. It's up to the employer whether to offer it or not.