May depend on the kind of transaction: a contract for deed, or a deed with purchase money mortgage. In the former case you are not the owner, or are not entitled to a deed, until a certain point in time and you would need a local attorney to advise you on the nuances of that situation. If you buy, get a deed and give the seller a promissory note and mortgage, you really own the house, and if the seller dies the note and mortgage become the seller's probate estate assets and nothing needs to be said about it in the contract. But often sellers "discount" such notes asap after a sale, and sell them to someone, for example if you would owe $100,000 and are good credit risks, the seller may be able to find a buyer for the NOTE and grab $90,000 in cash today for it, but that would NOT change your rights and obligations under the note or mortgage.
As to improvements, you would normally be required to notify, and potentially get the consent, of whoever held the "paper" either way - you'd have to look at the paperwork and see. Florida generally treats obligates that are "tied" together like a note and mortgage to be a mortgage, even if the transaction appears to be just a contract. Tax considerations are separate but may depend on whether you are responsible for paying them to the seller, or to the county. As to possible homestead issues as they relate to bankruptcy, bankruptcy laws tend to follow local law as to what the nature of a transaction is, so you're back to the "tied" situation. Contact a local attorney to help you.
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To address your questions: If the seller gives you a Deed and takes back a Promissory Note, then you own the home and the seller essentially retains a secured "I.O.U.". Via this method, I recommend you use an attorney as your title company, since typically an attorney will not charge attorney's fees and merely act as title agent for the transaction (that's what I do). It's like getting a free attorney. In this scenario, If the Seller dies, then the Promissory Note will be inherited/devised through their estate (e.g., via their Will). You own the home, however, and can claim it as your homestead and make improvements, depending on the terms of the mortgage.
Alternatively, if you are considering a Contract For Deed scenario, I highly recommend that you retain an attorney to do a title search and look into title insurance on the Contract For Deed. I recently encountered the scenario where there were tax liens on property my client bought via Contract For Deed but have not been cleared; therefore, there is a chance they could lose the property.