A couple notes.
1) Depreciation schedule is typically set by the original purchase price. Seems like the only "lower of current value or purchase price" situation you mentioned is if you lived in the property before turning it into a rental. When it converts to the rental, you take the lower of the two values, calculate the depreciation and it is fixed at that amount moving forward. As you mentioned, 'Improvements' divided by 27.5 is how much you depreciate per year. Once a deprecation schedule is started, it should not change year to year. Certain things reset the depreciation like death of the owner. There is a "stepped up basis" to the full market value on the day they died and you can restart depreciation from the new value. Just refinancing a mortgage does not change your depreciation schedule. Talk to a professional to see if buying out a co-owner does anything, I don't know.
2) You do not need to go off of the tax record to decide your 'Improvement' value to depreciate. Looks like the way it is calculated varies county to county, state to state. For my properties, the assessor always puts the values at 50% land/50% improvements. You need to decide the fair breakout between land and improvements and use that. Be realistic, use numbers you can justify to the IRS if they come calling. For example, I bought a $1M 4-plex. Assessor listed it as $500k Land / $500k Improvements. After talking with my tax adviser and looking at comparable real estate for the area, we agreed that $300k Land / $700k Improvements was a fair valuation and justifiable. $700k / 27.5 = $25,454 per year in depreciation. That should not change for as long as I own the property.