There's no such thing as an "appraised" rent value when it comes to taxes. Either you're renting at or above FMRV, or you're not. Period. The FMRV is basically, what a renter is willing to pay. The IRS looks at it as compared to rents charged for comparable houses in the same general neighborhood.
When you're renting at below FMRV, then you are considered to not be renting for profit. Therefore, your deductions are limited to the income and may not exceed that. So no losses can be claimed. Keep in mind also that the period of time to seller is living in it after your closing date of the sale is "unqualified use", and this will matter if/when you sell the property down the road and the "2 of last 5" rule comes into play.
All of this is covered is IR Pub 527 at https://www.irs.gov/pub/irs-pdf/p527.pdf
When renting for less than 1 year you are not required to depreciate the property, and I suggest you not depreciate. Otherwise, it's something you have to track for the entire time you own the property.
Keep in mind that when classified as a rental, only the mortgage interest portion of the mortgage payment is deductible, and that has to be pro-rated for the period of time the property was not classified as a rental. The principle part of the mortgage payment is never deductible. If you pay for mortgage insurance (different from property insurance) then that may or may not be deductible (prorated of course) depending on your specific circumstances. The program will "Know" based on the data you enter.
Homeowner's insurance is not deductible as an itemized deduction on SCH A. However, you can deduct a prorated amount on SCH E for the period of time it was a rental.
There is no such thing as "start up expenses" for residential rental property reported on SCH E.
Overall, you're better off in the long run to just report the income as miscellaneous income and not deal with the SCH E, since you're only talking about 5, maybe 6 months here. This way, the only thing you have to deal with in the future, is the 5-6 months of unqualified use when the property was not your primary residence.
