Given metal dimensions and material, is it possible to determine the maximum current that the structure can handle? [semiconductor processing]
Well, there are papers out there such as this one that predict the fusing current limits. The current limits set for normal long-term continous operation of ICs are not limited by fusing current, but by electromigration, so they'd be pretty conservative.  

You could also just sacrifice a few samples to determine what the limits are empirically. 

I'm not sure this will lead to the results you want- whether it's aluminum or copper or something else, the metal is going to have a significant temperature coefficient, and if you're pulsing it so it heats up by hundreds of K, how will you separate the temperature effect from the room-temperature resistance? I suppose if you had enough points you could extrapolate a curve back into the grass where the temperature change is minimal. 



I know every problem doesn't have a circuit design as the solution, but could you just make a preamplifier for your tester?