Checking Vref is a valid way of ensuring that the weight in your Performance Tool is the same weight that the FMC is using.
1) What's the rationale for using less than 100% power at take off? Other than cost, is there something else?
Not to my knowledge. It all comes down to cost. It actually burns more fuel to take off at less than full thrust, but the savings related to engine wear outweigh this. It is totally up to the pilots to decide when to use full thrust. An example might be when there is a gusty cross wind or windshear reported.
2) Given the plane knows the start and end destination in its avionics, couldn't it at least alert a weight that would seem way out of the norms when input?
More often than not the weight in the FMC is correct, it is the weight in the laptop or IPad that is incorrect and not in any way integrated with the aircraft so no warning is possible.
3) Given the avionics know runway length, pre-set power, and V1 couldn't the avionics suggest an abort of takeoff if the performance characteristics seemed significantly out of bounds?
FMC may not know the runway length, does not know pre-set power, and does not know V1. These are entered by the pilots after doing calculations on an iPad or a laptop or manually from a book.