...Yet so limited?
Not nearly as limited as you seem to believe.
Can we measure it?
Sure. In many different ways even, from complexity, to weight, to energy usage, to which parts are linked to which other parts.
E.g.: How many grams does consciousness weigh?
The machine that computes is weighs about 1300 to 1400 grams. Asking how much the actual processing weighs is like asking how much does a bitcoin blockchain weighs. It's software/information, being processed by a biological machine.
How much energy does it have?
Less than 80 Watts when dormant, and approximately 100 watts when doing normal everyday activities.
During delicate surgical procedures (or epic parties), can it be safely stored in a freezer (or some other non-human vessel) and put back in later?
Can we transplant it from one person to another? Or across the species barrier?
Theoretically, yes. We just don't have the technology yet, due to its complexity. To get a sense of what a brain is, imagine a huge cluster of tiny computers (neurons) all networked together in some specific way, where every packet sent from a source ends up passing through a bunch of other specific computers to its destination. The network cables get moved around as the network develops. So, let's say you want to think of a bear. Instead of just pulling up a data file on bears, a packet gets sent through the system, which passes through computers that store information on topics like: furry, brown, 4 legs, round ears, snout, teeth, claws, forest, dangerous, etc. As the packet travels through the system, it travels through specific network nodes and actives all the things that are related to "bear" that are deemed most important to the concept, allowing us to come to a general concept of a bear. That's how all information and all thought is stored and processed by our brains. In order for us to safely store that consciousness in a freezer, we would have to figure out how to freeze the wet network without the expanding water molecules (water expands when frozen) tearing the network up. We may figure out how to store brains safely without having to freeze them, or freeze them by some other method. As for transplanting, if we are able to scan the network structure and rebuild it from scratch, building a cluster with the same connections, then we can duplicate /transplant consciousness elsewhere. Likewise, if we are able to successfully disconnect and reconnect all the tiny nerve endings between the brain and the spinal column, then we can transplant brains. It's not a question of how, it's a question of do we have the technology.
Seems to me that there's practically zero evidence that it even exists in the "physical world" (zero mass and energy?!), yet it somehow exists.
LOL! Yeah, there is exactly the same amount of evidence as for the operating system running on your computer and the browser you're using to read this.