Help caliper won't spring back...flumoxed.

Have you tried taking the spring off and bending it outwards to provide more resistance? With the spring unhitched, try working the caliper open and closed with your hands. Is it stiff? The fore and aft nuts sometimes get too tight. Or the bolt could be bent. Presumably you've done the obvious and sprayed some lube (WD40 or similar) into the joins along the length of the bolt? Finally, check the brake lever to see that it's not catching or stiff.

I'd say check for kinks/bends in the cable outer and inner, but if you've put a new one on that shouldn't be a problem. Unless the routing goes around a sharp corner and you're giving the new cable the same problem as the old one.

You don't have the quick release lever in the open position do you?

The full outer shouldn't be a problem, unless it had a lot of sticky grease inside. It should work better than a bare wire.

My first thought was cable routing so the obvious question, is the spring working properly? If you open the caliper using your fingers does it spring to the open position?
Have you stripped the brake assembly and tried putting more tension into the spring?
I know some calipers, Campags(?) have adjuster screws but I don't know about the R556.