1) Yes, a new HVAC system is a residential rental real estate improvement asset. In TurboTax under Assets/Depreciation, choose Rental Real Estate Property >> Residential Rental Real Estate. As an improvement, the cost will be depreciated over the same useful life as the residence itself (27.5 years, same amount each year).
Here is a table from IRS Pub 527 Repairs and Improvements showing HVAC is an improvement.

2) and 3) Rental residential real estate assets are not eligible for the Section 179 deduction or Special "Bonus" Depreciation. Appliances, carpet, and furniture for rental properties are eligible for these accelerated depreciation options.
However, you may be able to write off the cost of the HVAC if you qualify for the Safe Harbor Election for Small Taxpayers. TurboTax will ask you about this election at the beginning of the Assets/Depreciation section for your Rental Property (choose to review your answers to the annual elections questions). If you choose this election, you can expense up to $10,000 in costs that would otherwise be depreciated over a very long time. Be sure to read any blue links "Learn More" to be sure you can take advantage of this election.
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