This is why outpatient therapy can be... ineffective. It's like the circuit between wanting to do something and getting it done is fritzed. It's not lassitude or laziness, it's just a brain fault.
Is there any chance of getting in-home outpatient speech, OT and PT? I know getting out to go to therapy is part of the therapy, but it's the balance between compliance with therapy and the additional benefits of excursions.
In terms or hardware for AC: this is a place where a Nest thermostat might be a good idea, since it can be monitored from anywhere via an app. It requires a live wifi connection in the house, but they're simple to operate (much more so than the button on keypad type.)
Certainly, her doc needs to be in this loop, and if you have an elder care social work system, getting them in the loop will help. (Fresh eyes and all that, plus familiarity with services like Meals on Wheels, senior day programs, and care resources.)
As for the boyfriend... I agree, he can continue to pay his half of the base, but heat cannot be helping. Does the power system have a budget plan, that averages out the bill over the year? If they're already on that, well, the uptick won't come until whenever the re average is, and if not, switching them to that might cool his jets a little.
Also, that's two people in a fragile state with one as primary on-site caregiver. Personally, I'd be extremely wary. Even if he's fine, he's um... Past warranty and caregiving takes a physical toll. Social work definitely needs to be monitoring, even if only dropping in occasionally for welfare checks.