If you live in the home for more than 2 years, after getting title, you will be able to exclude (up to $250,000) the portion of the gain attributable to residence time.
You will have to allocate the gain between the time it was YOUR rental and your residence time. Apparently it was your rental 2010 to 2012, but not 2013 to 2021.
You will have to recapture all depreciation "allowed or allowable" from 2010 to 2021. As others have said, your sister's basis, including depreciation transfers to you with the gift.
Your cost basis is what you and your sister paid back in 2010 (~$200K). The $500K present value is not relevant for income tax purposes (but is relevant for your sister's gift tax* return)
That's my opinion, based on the facts as I understand them. As others have said, professional help is advised.
*"Gift Tax" is somewhat of a misnomer. Even though a gift tax return may be required, very few people ever actually pay federal gift tax. The purpose of the gift tax return is usually only to document a reduction in the allowable estate tax exemption.
See https://turbotax.intuit.com/tax-tools/tax-tips/Tax-Planning-and-Checklists/The-Gift-Tax-Made-Simple/....
