Yes, yes, and almost certainly.
Unless there is a specifically noted "peak current" spec that is higher than the "rated current" spec, you have to assume that the peak current cannot the maximum rated current. 5 seconds is long enough that it can generally be considered "continuous" anyway.
It seems pretty clear that pushing 600A (let alone 1200A) through a connector rated for 300A is way out of spec. Of course, being out of spec doesn't guarantee that the connector will fail... but I wouldn't count on it working.
Also, take a look at the opening time curves on your fuse. Your 600A fuse will take 10 seconds to blow at about 2400 (!!!!) amps. This is typical - fuses never have nice, sharp cutoffs - there is a large range between their rated (continuous) current and the current that opens them quickly.
I don't know how much current your supply is capable of, but if a short circuit produces anywhere between 350A and 2400A, I'd bet on the connector blowing long before the fuse. If your short circuit produces upwards of 4000A, your fuse will blow in around 10mS - but then, your connector will blow much faster as well, so the connector could very easily still blow first.