I purchased my sister’s home and paid it off allowing her to stay as long as she wants for approx $350 a month. Can I depreciate and claim losses on the house?

"Can I depreciate and claim losses on the house?"

Not unless your sister uses the home as her main home and $350 per month is considered to be a fair rental price.

See https://www.irs.gov/taxtopics/tc415

Otherwise, each day you rent to her at below fair rental value is considered a day of personal use.

Since you are renting to family at well below FMRV, expenses that exceed the rental income are not deductible and you can not carry them forward. You just "lose" them permanently and forever.

At $350/mo you have $4,200 a year of miscellaneous unearned income. I wouldn't bother converting this to a rental at all. Just report the income for what it is and pay your taxes on it. an extra $4,200 a year isn't going to make that much of a difference in your tax liability anyway, and it's not worth the effort of dealing with this as rental income anyway.

Besides, since rental property has to be depreciated, the depreciation recapture when you sell the property in the future will have a huge negative impact on your tax liability in the year you sell. So it's not worth the effort.