No problem at all, the built-in compatibility mode will conveniently turn your rechargeables into disposables.
Jokes aside, while the technology and the "why did nobody think of this before" is certainly impressive, if this gets any market traction it will cause more landfill by hurting rechargeable adoption than it will prevent by slowing the disposable battery replacement cycle.
According to the article Roohparvar says the sleeve will also work with 1.2 volt rechargeable batteries, boosting them up to 1.5 volts., but they don't say anything about how well the rechargables would handle this usage.
The usual way for rechargables, I imagine. They'll handle it just fine until they die from an overly deep discharge.
If the chip is very clever, then it'll detect that it's working on a rechargable (from voltage curves?), and stop discharging after a certain point. But the specifics there depend on the exact chemistry used, so it'd be difficult to get it right.
Definitely yes, but (example) the rear light of my bicycle insists to be powered by two AA 1.5 V batteries. Two rechargeable ones don't have enough voltage to light it so I need disposable batteries. The front light works with 3 rechargeable ones. We should stop building stuff that needs throw away batteries.
I probably change the batteries every month or so, which is usually about 40 hours of riding. Probably too often, but whatever. The light cost $15, I think.
(Actually I'm retiring this light today and switching to a dynamo-based lighting system. No more recharging ever. Looking forward to that.)