I just finished the basement for my rental property. Is there a way to use bonus depreciation for it?

No.  A residential rental property itself does not qualify for bonus depreciation.  Personal property you use in the rental, like refrigerators, dishwashers, furniture, etc. do qualify.  But not the building and improvements to it.

In TurboTax if I add the basement as a property improvement and depreciate it over 27.5 years, can I add the appliances, furniture, carpet, etc. as a separate improvement to depreciate it over 5 years?

Yes, you can add the other additions as rental assets. These are not part of the "improvement." TurboTax will use the correct class life based on the asset category you choose. Appliances, furniture, and carpet fall under the same category.