That was OK back in the day - but not recommended for modern cars. The fuel pump in our cars is an immersion type - meaning that it is cooled and lubricated by gasoline. Driving to point of almost running out of gas will really shorten the life of these pumps - hence the reason why there is a generous reserve in the tank, even when the gauge indicates empty.
Same way with premium gas - in most cases, it doesn't contain any more or less additives than regular gasoline, sometimes the refiner will "cheat" and add more alcohol to boost octane rating. Won't make the car run any better, but the alcohol can absorb moisture that is in the bottom of the tank, which could be beneficial - which is where this urban legend of sorts, probably got started.
As for the stumbling when you step on the accelerator - I'd say double check that the throttle body is not gummed up, has a lot of varnish/carbon buildup. Double check that the PCV valve is clean and working properly, double check for vacuum leaks - especially if you recently Seafoamed the car - real easy to introduce a vacuum leak or forget to put a vacuum hose back on after Seafoam treatment.
Also this being a 2000 Corolla? Check and clean the MAF/IAT sensor. These are pretty easy to get fouled, and many have found great results by just cleaning them. Check online for MAF cleaning - lots of DIY guides out there for reference. There should also be one on this forum as well.
Since the longer the car is sitting, the harder it becomes to start - could be related to a leaky injector or oil fouling from leaking valve seals. Do you notice a puff of smoke after the successfully fires up? If there is, what color - bluish white smoke, black soot like smoke, white smoke, any odors?
Plugs - stick with OEM style iridiums, this engine seems to be pretty picky about plugs. Double check chassis grounds - a ground loop can cause problems down the road, as the ECM is very sensitive to noise.
Okay, I can live with that being an urban legend. I heard that if your car wasn't made for the higher stuff, spening the cash on the higher octane won't matter. Is that true or somewhat true? Not that this question matters all that much to me at the moment...
The PCV value, MAF and the what now? I don't notice any odd smells, or smoke. I stayed with the original plugs. Im not sure what the chassis ground is.. I thought the chassis was grounded through the tires?
Ignorance is bliss at times. Ill darn darned if I stay in the dark. Feed me your brain, fishexpo101! Pronto!
...here's a tablespopon, shiny and silver.