Modern Vespa

I live in a very hilly area. It's a river valley surrounded by canyons, deep gulleys and rolling hills. The most extreme hills in the immediate area are 2+ miles long and 15%, at the steepest. Most hills are in the 5-8% grade. Last summer it got as hot as 120 degrees F, but that is exceptionally hot for here. 95-100 degrees F are more typical summer highs.
Assuming 180-220 lbs rider/cargo, can the stock GT200 and ET4 150 cooling systems handle:
a 2 mile long 12-15% grade hill at 60 mph?
a 2 mile long 10-12% grade hill at 30 mph?
steep rolling hills at a crawl to 15 mph?
idling when parked in the sun?

Yes.

Keep up on your maintenance, and keep the radiators clean.

The difference between 'normal' ambient and 120F is trivial compared to the coolant's 'hard riding' operating temperature of just above 212F. Yes, the fan may come on if you're going less than 30mph, but it's designed for that. Italy also gets hot and some hills there are 20% or more!

I'd be more worried about keeping the rider cool enough in those temps. Lots of water, half inside and half soaking the T-shirt, plus a 'cooling collar'.

The scooter doesn't really know if the load is coming from air resistance or gravity.

What about the ET4 and being air-cooled?

The air is sucked in from a fan on the crankshaft, it's independent of vehicle speed.

Ok. What about the air-cooled ET4?
Good point about Italy's hills! Does anywhere in Europe get that hot?

True. I was thinking about air flowing over the cooling systems. I know the GT200 is water cooled and has a fan. What about the air cooled ET4? Does it have a fan?

Oh! Is that for the engine? I thought that only cooled the belt and clutch.

Nope. Sucks the air in and then over the cylinder cooling fins. That's why the cylinder has a shroud around it.

Both Italy and Greece regularly get temperatures above 45C (113F) on occasion. Recent records have been 118F to 119F - each year getting closer to 120F. Tourists especially have been dying from heatstroke just this last week.

I rode the Italian alps in my 125cc. At points during the climb I was full throttle 23mph

These engines / transmissions cope absolutely fine with this.

For a bit of mechanical sympathy, I personally would check oil levels are good before any heavy duty stuff.

Also, do not shut off the engine immediately after the climb. Give the engine a mile or so of easier work, or even idling for a bit to bring the temps down before you switch it off.

Oh, good to know!

Ok. I had no idea! That make Italy being in Africa in WW2 make more sense!

Good tips, thanks!

Athens in Greece and Death Valley in California are very close in latitude...