This should be a simple problem but I just can't seem to figure it out. When I test the resistance between the power and ground leads of the "NeoPixel Ring - 24 x 5050 RGB LED with Integrated Drivers" I get an overload (i.e. there's no connection, very large resistance) which we would expect. However when I connect two of them in parallel I get a resistance of the pair of 1.4K Ohms. I have nothing else connected to them (i.e. the signal leads are not connected.) It makes no sense to me.
Ultimately I'm trying to power them with small CR2032 batteries, both they can't power the rings with this semi-short situation.
Measuring resistance of digital electronics is meaningless; they cannot be modeled by ohm's law (ie, the I/V curve is highly non-linear). That is also nothing near a short. WS2812 LEDs pull a non-negligible amount of current while idle (in the ballpark of 1mA IIRC); they weren't designed for low power applications. This standby current alone probably exceeds what a CR2032 can comfortably supply for 48 LEDs - and you haven't even turned them on!
Your project will not work with a CR2032. A WS2812 LED turned on can pull up to 60mA per LED, so your 48 LEDs will pull up to 2980mA. A CR2032 battery is good for like 20-30mA. You need a battery capable of supplying approximately 100 times as much current as the coin cell you have selected.
Depends how bright you've got them on. Certainly not at full white brightness. You can search for the size of battery, word "datasheet" and the manufacturer, and they give capacity specs at various loads.
At full brightness, you're pulling 3A, which will rapidly deplete even a pack of AA's - like 15-20 mins it looks like? That current is higher than the highest current spec they give, so it may well be even worse than that.
But if they're running at well under full brightness, you might have a chance. You'll need to measure current draw with the patterns you want.