To start with, the property is considered rental property because it was not your primary residence for 2 out of the last 5 years.
Improvements should have been amortized over time, just like the building should have been. If that was the case those improvements would be already included in your basis, and the depreciation would have been accounted for.
If you simply took the expense for the improvements and took the deduction in the year that it occurred, you would not include it in the cost basis. You've already taken the full deduction.
Yes, you can add allowable closing costs to the cost basis for both when you purchased the home as well as when you sold it.
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