However, I don't know what is too hot for the processor, so could someone inform me what the danger temperature is?
Generally, your PC will cut out before any real damage can be done. The only genuine danger I can think of offhand is if the CPU cooler has become unseated, or even slightly so. If it's a bit wobbly, not secure, then make sure it is secure again, otherwise yeah, your CPU could perhaps die.
I doubt that's the case though. Things might be going a bit weird, if so.
Any particular reason you've taken to watching CPU temps? Advice from someone/somewhere? Just seemed lika a good idea at the time?
Regardless, ~60C is not extreme. If you want to know the tolerance of a given model, go to the website and look it up. In your case, intel.com of course.
Anyway. What are the temps like @ idle? Oh, and I assume you're not using aftermarket cooling and thermal compound, right?
As a point of reference, my e8400 (E0 stepping, aka the good stuff) idles below 30C, often around the 27-28 mark. But then, I do use the two things I mentioned above. Plus, a nice case. And, what's more, the e8xxx series is a die-shrink and minor upgrade of what you're using (e6xxx) - and therefore, running cooler. And I have automatic speed reduction enabled via the BIOS, so it's most often running at about 2/3 its standard (overclocked) speed. At full load, it might hit 40C or so. But keep in mind the points of difference - it all adds up.
I heard Intel processors have a higher heat tolerance than AMD, which I used to have, but I don't know the specifics for this model:
Not necessarily. It's more like a chip by chip thing. Also, stepping, batch, model, range... each factor plays a part. Not to mention selection of aftermarket cooling and thermal compound (if any)... Basically, heat tolerance is not all it's cracked up to be. Running hot is not a great idea, essentially.