Usually private medical centers in the cities. But public hospitals are also catching up quickly. Most medical staff and doctors are trained for their sub-specialty in the U.S. and Europe. You get both worlds. Philippine General Hospital is more equipped and trained than Denver Public Health IMHO.
Nope. Healthcare in the Philippines is on par, if not greater than the US. And a lot cheaper too. I guess you have not been in the country for a long time.
For many years now, at least a decade now, they do not use "X-ray Films" anymore. I don't know of any center/hospital use them nowadays. Everything is digital now.
She will need a travel document or something like a boarding foil from the nearest Phil. consulate. Consulate in SFO used to issue those all done the same/next day for emergency trips before covid.