You will not need to worry about past depreciation on your inherited property. You will just use your stepped up basis (FMV of property on date of inheritance) and this new basis will be used for depreciation. You will be able to depreciation these inherited assets in full over the property's useful life. For example, use the full 27.5 year, S/L for the rental house (less land) and the start date will be the date when the rental property was transferred to you.
For any prior capital improvements, these will be included in the stepped up basis on the inherited property so do not depreciate them separately.
For any appliances, since they are considered "new" to you, you will just use the new FMV of these items and depreciate them over the new useful life at the date the asset were transferred to you.
