Yes, the date property tax is paid is when you can deduct it as an expense. You do not need to prorate.
Answer to question 1: If the property tax was paid after the property became available for rent, then you should include it. For tax purposes, it is always when it was paid and not the assessment period of the city on the tax bill.
Answer to question 2: Any tax bill paid in 2023 is an eligible expense deduction in 2023 as long as these taxes were not paid for arrears when the property was purchased. Based on your statement about a current reassessment, then it is deductible.
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