Not sure about definitive proof, but one could trace between a few directly-connected nodes i.e. along the pacific rim. Then, making the same assumptions as the article (i.e. cables lie along great circles, most of the measured time is spent in traversing the cable, etc), demonstrate that those connected nodes do not lie on a flat plane.

The more nodes one has mapped, (i.e. distance measured to surrounding nodes) the less likely one is able to orient those nodes into any shape other than a sphere.

I think yes. If we assume reliable conversion of traceroute results to implied distances, and can generate selected data (routes) of our choice. That is, we can see results of remote hops that aren't directly connected to us. Then, ultimately, the resulting matrix of results would (mathematically, ++) conform to a spheroidal shape and no other.

++ think Map Projections, and the associated math.