I don't know whether the method of acquisition gets reported to the IRS. As far as FMV on the date of the gift, I simply called the town's tax assessor's office and they provided that value to me. I don't know which IRS form or schedule this information appears on since I have not completed my taxes. However, our house was gifted to both me and my sister, split 50-50. She lives out of state so the closing attorney completed an IT-2663, Nonresident Real Property Estimated Income Tax Payment Form, to compute the tax she owed to the state. Nowhere on that form can you discern the property was gifted. Also, I tried both "gift" and "other" in TurboTax and the results on the "You Have a Long Term Gain from This Sale" screen is identical using both methods (if I use the same value for cost basis in both).
I encourage you to read IRS Publication 523, Selling Your Home. I went through the worksheets to calculate the gain from our sale. In short, if the donor's adjusted basis was less than or equal to the FMV at the time of the gift, your basis will depend on WHEN you received the home. There are 2 short worksheets for homes received before 1977 or homes received in 1977 or later. Both worksheets require the FMV of the property at the time of the gift. The result from the worksheet is your cost basis at the time of the gift. However, as you complete all the worksheets, you ADD improvements that were made to the home since the gift date to calculate your Total Basis at the time of the sale.
In my opinion, TurboTax should add an additional screen for gifted property to allow you to add improvements after the gift date.