L'Hopital's rule allows one to evaluate x*log(x) as real x approaches zero from the positive side. The limit turns out to be zero even with slight modification of each term, (eg, replacing the product with (ax)*log(bx) still leaves the limit as zero).
Without context, that's an arbitrary choice of a limit, albeit an aesthetically pleasing one. Generally speaking there is no reason why 0log0=0 makes any more sense than 0/0=0. Either might make sense to use as convention for a specific problem, which again is a temporary local modification to notation.
There is an identity rule for division: anything divided by one is itself (x/1 = x) but there is no rule that says x/x = 1 You can derive "rule two" from the identity rule for multiplication x*1 = x --> x/x = 1
Uh, that's not true. You can't derive existence of multiplicative inverse from existence of multiplicative unity, you have to assume existence of multiplicative inverse. For example, the ring of integers contains multiplicative unity, but does not contain multiplicative inverse. And there is no identity rule for division, there is just an identity rule for multiplication -- i.e., 1 is defined as the element that has x*1=x for all x.