Homeowners during the Redemption Period

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I have a few questions about homeowners and evictions after the auction

If a property is purchased at auction, what rights do the homeowner have to the property during the redemption period? In other words, does the new homeowner have the legal right to have the homeowners evicted as soon as the property is paid for and the title is transferred?

Has anybody ever "paid" off a homeowner to leave via a written agreement rather than have to pay for the eviction process?

How long will homeowners typically try to stay in a house if it's going through foreclosure?

Thank you.