No, you would not change the cost basis of the house because you had an appraisal or refinanced the house. If you first rented the house in 2022, then you would use the lower of cost (what you paid for it plus cost of improvements) or Fair Market Value (what it was worth when you converted it to a rental) for your depreciation basis. So, if you what you paid plus improvements you made is less than the current price for the house, you would use the price you paid when you bought it.
If you rented it out prior to 2022, you would not make any changes to the depreciation basis that you were already using. Also, if you took cash out of your rental, and used it for things not related to the rental, you would not be able to deduct the interest on that portion of the loan. You will need to prorate the interest deduction between rental and whatever the cash was used for.
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