Jack oil mixes very quickly with use. Just jack it up and release it a few times and it will be mixed. You are the pump.
A hardened, shrunken seal may soften and re-swell. A damaged or worn seal is lost.
If you have a mineral oil based jack, and most are, you can use, ATF, motor oil, gear oil, power steering fluid, hydraulic fluid, or pump oil. I routinely use a mix of DexIV and STP as jack oil. I use the Dex to make sure that entrapped air is released quickly and the the STP to purposely increase the viscosity. The thickened fluid provides for a very smooth release rate. I have not had much problem in the winter, but I do not use floor jacks often in the winter.
I have twice seen leaks start fairly quickly after use of RedLine power steering fluid as jack oil. I don't know if it was coincidence or causative.