Question: The heat proved too much for a few people who'd gone off too fast for their capabilities and ended up suffering from exhaustion.
Does prove mean this to show a particular quality after a period of time like in this example The new treatment has proved to be very effective or this is a special phrase which you should use with too much?
end up doing something What does it mean? I read the definition in the dictionary but still can't get what it means when we use with the ing form. Does it have only one meaning?

Answer: 
"Proved" or "proven" in these contexts means "turned out to be". 

There is an initial unknown situation (how would the runners fare in the heat? How would the new treatment go?), and as the situation is played out, the result becomes clear - the result is proven, like a scientific experiment. That is what the "end up" refers to - at the end of the situation, the initial "question" is answered.

It is fine to use it in everyday speech, it might be a *tiny* bit more formal than common, everyday language in the 21st century, but it is still very accepted.

You don't need to specify time using "ended up" or similar, and you don't need to use it with "too much", it's fine by itself.
To prove something means to show it is true but can also mean “found to be” Adding ing to a word means you are doing that thing at the moment. So proving would mean you are currently in the act of showing truth.