Coaster brake heat management and lubrication I've heard that coaster brakes can reach high temperatures, so it desirable to use grease designed for high temperature applications.
My questions are:


Would a grease labeled max 220 °C be good enough? Or it can get much hotter? 
What happens if the grease is not good enough? Will it liquefy and drip out of the hub?
How should you use the brake when going downhill to avoid overheating? Is it better to use it in short bursts, or continuous but not very hard braking?
How hot can it get exactly? A chart showing the heating and cool down rates would be interesting, but I don't think anybody has done the experiment, it's hard to stick a thermometer in the hub (maybe doable with a temperature camera sensor?).
Most people who aren't using a manufacturer grease are using a high temp automotive grease. I'd suggest looking at the manufacturer's recommendations and following them; brakes aren't a place to save a few bucks. 220 C seems low for the type of grease; people have failures with high temp automotive greases which often rated for 100 C+ higher than that on tough descents. 
Brake fade, loss of braking. Brake might be smoking and ruined. Loss of grease and dropping of the grease is the least of your problems at that point given the fact that you're probably in for a not so nice crash. 
You need to give the brake adequate time to cool off. This means controlling your speed by modulating your brakes (use in controlled bursts). You should also have 2+ independent brake systems which you can alternate with. If you use continuous braking over a long descent, you're just going to build up heat and have brake fade. Unfortunately, coaster brakes are prone to this in long descents making them not ideal; my advice is not to take a coaster brake bike on long descents. All brake systems will fail if overused.
Pretty damn hot ; an old article but probably still quite relevant. You normally only see coaster brakes on kids bikes and bikes which are used in low load leisure systems (e.g. beach cruisers), where this would not be a problem (not going very fast, not braking very hard for prolonged periods of time).  See (1). Measuring this on the surface of the hub wouldn't be too hard (thermal camera), but internally, you'd need some work.