Inherited Property

I have several questions regarding inherited property:
1) I have appraisal of a townhome which will be used as cost basis as of date of death. This was rental property which I am now choosing to sell in 2023 vs keep renting and was vacant when my father passed away so I never rented myself. My father passed in 8/2022 and property was transferred to me in 11/2022. I made improvements to the property in 2023 before listing for sale in 2023 (new carpeting, replaced shower fixture in bathroom, and misc minor repairs/painting where my husband & I performed the labor ourselves. Can I add the costs of improvements regardless of value to my cost basis (adjusted cost basis)? Can I calculate my labor cost and add to cost basis? If so what is acceptable calculation for my labor cost?
2) When I sell the townhome in 2023 I will report on my 2023 tax return. How will I treat the closing costs I incur when I am calculating the capital gain or loss? I am using a realtor so the closing costs will include commission as well as various fees, property tax, doc stamps, etc. Also, how do I treat the monthly expenses I incur such as utilities and HOA fees until the property is sold?
3) For 2022 the property was rented Jan through April 2022. The tenant passed away and smoked in the townhome. Therefore, the townhome could not be rented again without major cleaning and repairs. The intention was to get the townhome in condition to rent again, however that never happened before my father passed away in early August. I will be filing final tax return for my father which will include the rental income. How many months can he reflect the townhome as rental property? Not sure what the stipulations are around months where property is not rented due to repairs needed?
4) The townhome was transferred to me in November 2022 and still needed major cleaning & repairs to either rent or sell and was vacant. How do I treat this property on my 2022 tax return? I did not make any income on the property in 2022 but incurred expenses such as annual property tax, HOA fees, and utilities in 2022. I made no improvements in 2022. I made decision in January 2023 to sell vs keep as rental property.
5) My father's homestead property remained deeded in his trust for all of 2022. Even though he passed in August can I still deduct the property taxes that were paid in November 2022 on his 2022 tax return?
6) My father's homestead property will be transferred to me in 2023 and remains vacant. It will take me some time to go through all the belongings and decide what to do with the property. How will I treat the property on my 2023 tax return and the expenses associated with it such as insurance, property taxes, utilities, repairs, etc.?