Temperature compensation seems to work very well here, although I haven't done any extreme tests. I have a couple gadgets with DS3231s sitting on window sills. They catch direct sun and in the winter probably are cooler than the rest of the house. They've been operating continuously since last fall sometime and are both within 5-10 seconds. Like I said, that's not an extreme test, but I've tried non-temperature compensated RTCs in those same locations and they drift a lot farther than the DS3231s, and also a lot farther than non-temp compensated RTCs in other parts of the house where temperature is more constant.
I use regular lithium coin cells for all my RTCs, including DS3231 and DS3232. I use CR2016 mostly, but also some CR1220 (which may be a bit on the wimpy side, but should still last for years). I hardly see the point in a rechargeable cell, estimates I've seen indicate that the regular lithium coin cells should last 5-7 years.