For depreciation purposes, your cost basis is the "lesser" of the FMV on the date it was placed "in service", or what you paid for it when originally purchased. Typically, what you paid for it, plus the cost of improvements will be lower than the FMV on the inservice date, plus the cost of improvements. Do not add the cost improvements to the FMV on the in service date if the FMV on the in service date already includes those improvements. So whichever is lower is the value you must use for depreciation. Either:
Original purchase price plus property improvements or;
FMV on in service date, with improvement costs added to it only if that FMV does not already include those improvement costs.
If I understand and interpret your post correctly, your cost basis will be (must me) what you originally paid for it, plus the cost of all property improvements, since that is the lesser amount.
If you lived in the property for the 5 years prior to it being a rental, then I would think it was your primary residence and you would qualify for the "2 of last 5" capital gains tax exclusion (if sold at a gain). Otherwise, if you treat it as a 2nd home then you get no exclusion. (I don't think this is your case though since you sold at a loss.)
treat it as a rental property for the full 10 months, both years.
You really don't have a choice in this. Since it "was" "in fact" rental property for ten months and those months cross a calendar year, you kinda sorta "have to" treat it for what it was.
All expenses incurred starting in Sept 2019 are rental expenses for the entire time you owned it after that.
Also understand that taxation on at least "some" of your depreciation recapture may be offset if you sold at a loss.
Remember, when selling the property the cost basis for the sale is what you paid for it, plus the cost of property improvement, *MINUS* all prior depreciation taken.
