The basis of an inherited home is generally the Fair Market Value (FMV) of the property at the date of the individual's death (or the alternate date if selected). If no appraisal was done at that time, you will need to engage the help of a real estate professional to provide the FMV for you. There is no other way to determine your basis for the property.
Yes you can e-file an IRS 1041 (estate return). After you install TurboTax Business and begin working on your return, you'll be asked which type of return you need to prepare. Select Trust or Estate return (Form 1041) and proceed.
If you were asking if you can e-file your individual return, I have not read anything in your question that would prohibit you from e-filing your individual return.
