Weighing in on the absorbing-cold question:
"Absorbing cold" is equivalent to shunting heat--rather than sucking the cold out of your target, you're pushing your body heat into it. Leaving aside the logistics of how you're pushing the heat, this could work rather nicely, since thermodynamics tells us that pushing heat from one object into another object of equal or greater temperature requires energy expenditure (thus, your refrigerator needs electricity to shuttle heat out of your frozen food into your room-temperature air). So, your character could "absorb cold" from basically anything, but the warmer it was, the more taxing it would be on her.
I like that idea a whole a lot. I'd want to keep the idea that the environment or object has to have a certain degree of temperature difference since that's a good limiter, but as the character reaches the limits of what they can get cold from, that could also provide a bit of extra drawback and drain.
But since you've answered my questions in such a way as to make me understand your core meaning, I can now safely tell my brain to shut up about this. And I would now like to offer my apologies; when you said "absorb cold", I never thought to compare it to the stone bench in my backyard. Had I done so in the first place, I would never have questioned it.
Alright, fair enough. Just an error in communication!