Heat has two problems for helicopters and airplanes that combined cause much reduced performance.
Simplified: Hot air is less dense meaning we get less lift at the same power/speed. And hot air reduces engine performance, making it harder to get enough power.
Helicopters can land at airports with much higher weight (or less power) than the limits for hovering and vertical takeoff. Because forward speed when taking off from a runway creates additional lift.
Emergency helicopters, like commercial transport, will also consider the risks of engine failure. You don't want to put the helicopter in a position where engine failure is unrecoverable, it's too high risk. Just like (commercial) airplanes don't take off from runways too short to safely handle an engine failure.
Hot air also makes air cooling across the cylinders (EDIT: for piston aircraft) less effective, causing the pilot to have to use lower power settings to keep CHTs and oil temps in line with the airplane manufacturer's guidance. I blow through 400F regularly on takeoff on hot summer days, while I can consistently keep CHTs below that in the winter.
I don't think any emergency medical helicopters in the Western US still have cylinders. They switched to turbine engine models years ago. But of course your general point about engine cooling still applies.
I seem to recall Phoenix closing Sky Harbor when the temperature went above (IIRC) 117 F. Fixed-wing planes generate less lift, too. (Or was that because the asphalt runways were softening? I think it was lift, but I'm not 100% sure.)
Phoenix didn't close Sky Harbor due to high temperatures. Some airlines canceled flights years ago when temperatures exceed 118 °F but those limits have since been raised.
There's no certification for heat per se; it's more that there's just a point where the aircraft can't take off safely from a given runway. Increased air temperature means increased takeoff distances for a given weight.
These are set by the manufacturer, and can be revisited as appropriate.
Not a certification limit, but you need valid performance data to be allowed to take off. So if the data is only provided up to 50 C / 122 F, then you can't take off legally at 128 F.
I'm very skeptical that helicopters can't safely operate at this temperature. It's likely a certification thing capped at the highest temperature they can actually test.
While there are plenty of certifications for aircraft maintenance and testing, this is legitimately a capacity issue. The air density issue is huge for aircraft. It caps maximum operating ceilings (as in the aircraft literally can't go any higher), it affects engine power (hence why things like intercoolers are an issue for performance engines), and it affects takeoff distance.
Ironically, it can increase the overall airspeed, since less dense air is easier to move through.
Simplified: Hot air is less dense meaning we get less lift at the same power/speed. And hot air reduces engine performance, making it harder to get enough power.
Helicopters can land at airports with much higher weight (or less power) than the limits for hovering and vertical takeoff. Because forward speed when taking off from a runway creates additional lift.
Emergency helicopters, like commercial transport, will also consider the risks of engine failure. You don't want to put the helicopter in a position where engine failure is unrecoverable, it's too high risk. Just like (commercial) airplanes don't take off from runways too short to safely handle an engine failure.