I work in real estate, this is how it usually works:
You negotiate for the house, generally if you are going through an agent, the buyer will sign an offer and put down earnest money (aka deposit) which they will lose if they back out of the contract. After you reach mutual acceptance, you sign a purchase and sale agreement. How this is done differs in various states. Some states have standard forms which combine offer and P&S, others the offer is different and you don't sign the purchase and sale until after other inspections are done. A lawyer can draw up a purchase & sale for you.
Generally at some point the buyer does a home inspection on the house, this is because of what htey call 'caveat emptor' which means buyer beware - the buyer has the responsibility to find anything wrong with the house that is 'discoverable' themselves. In general the buyer can't come back and sue the seller if they find something after closing, only in cases where the buyer had no way to discover the problem themselves (called a 'latent defect') AND they can prove the seller did know about the problem.
At the time you sign the purchase and sale you pick a closing date. This date could be tomorrow, it just depends on the terms of the contract, if the contract has a clause stating the buyer needs to obtain financing, then most likely the closing date will have to be at least 4 or more weeks out in order for the loan to get underwritten in time. Generally the seller pays all costs of maintaining hte house, their own mortgage up until closing, buyer pays all costs thereafter. Generally a closing agent makes sure that all the costs add up and everything is prorated and the money goes to the right person. (For example if you've paid the taxes for the year and are selling halfway through the year, then the buyer owes you the taxes for hte half of the year they own the house).
At the closing, title (or ownership) passes on that property. At the point when the buyer receives the deed to the property, you no longer own it. So you do not have the right to occupy it unless you have worked this out in advance with the buyer and have a written contract to that effect.
According the Statue of Frauds in order to be enforceable all real estate agreements MUST be in writing. This means that no verbal contract will be recognized by the court, so if you agree verbally with the buyer that you can stay on the premises another 3 weeks but do not have that in writing, you will not have a leg to stand on in court.
While if you have a buyer already you do not need a real estate agent, you definitely do not want to go through a real estate transaction without a lawyer representing you. The buyer should have one too, but of course you may or may not care if the buyer ends up with a bum deal... The lawyer can act as or get you in touch with an appropriate closing agent, help you secure title insurance, make sure you are paying applicable taxes, etc etc.
Oh the short answer - Yes, bank of america will want every cent for every days interest up until the day they get paid off.
