Yes, closing costs like you listed add to your basis for both original purchase and refinance.
As for the sale aspect of your question, I haven't sold any rental properties, so I don't know the mechanics of how to do it. I would add additional assets in TT for closing costs and improvements as non-depreciable assets like land. I would use the actual dates placed in service. This should not cause any problems like changing the original cost basis. As a general rule, I wouldn't change the original cost basis on any assets because the cost basis was previously submitted to the IRS on a previous return.
There are a number of ways to account for these additional costs. I hesitate to just add them to closing costs. If the costs are substantial, your total closing costs would appear to be too much compared to the sales price.
Again, I haven't sold any rental properties using TT. If @DianeW777 has experience in this area, I would defer to her judgement.
