The complexity of the puzzle game Net
Just a partial self-answer: I think the problem is NP-complete.

I found 3 gadgets (each one occupies $16 \times 16$ cells) that allows to build a Net game equivalent to a grid graph of degree $\leq 3$ and that should have a solution iif the original graph has an Hamiltonian cycle. The figure shows four different configurations of the gadget equivalent to a node with degree 3 (the full picture can be downloaded here).



The gadget SHOULD have the following properties (I'll try to check them with a constraint solver):


it can be linked to other gadgets only through the 3 interface pairs of corner wires ($A$, $B$, $C$);
the two wires of an interface pair must be directed both outwards or both inwards (otherwise there is an open ended wire or a cycle in the inner part of the gadget);
the gadget must be entered/exited exactly twice and from exactly two interface pairs (the green zones of the first three figures show the traversals A-C, B-C, A-B);
exactly two interface pairs must be directed outward (the red zone in the figure shows what happens if the three interface pairs are all directed outward);


The gadgets equivalent to nodes of degree 2 and 1 are similar (and we can also build "fill" gadgets to fill the holes of the original grid graph).

Now replacing the two central cells of one gadget with the central unit that sends the power on one direction and a terminal at the other endpoint, the game SHOULD have a solution iif the original graph has an Hamiltonian cycle.