Can the coil in a magnetic field be replaced by a copper mesh?
It sounds like your understanding of the theory is missing an important point:  Each turn contributes to the EMF.  There's a big difference between one turn of large wire, and 1000 adjacent turns.  For a large number of coils close together, Faraday's law of induction can be restated as

$$ \mathcal{E}= -N \frac{d\phi}{dt} $$

where \$N\$ is the number of turns, and \$\frac{d\phi}{dt}\$ is the rate of change of flux.  For simplicity, this is often developed using one loop. 

The wires used to wind the generator coils may look like plain copper wires, but they are actually coated in a thin layer of enamel, which electrically insulates them from adjacent wires. 

I believe that the mesh you show has un-enameled wire (Test for continuity between crossing wires to be sure) which connects at every crossing.  Using this to wind a generator would result in one turn, giving you an EMF \$N\$ times smaller than what you'd get with an equivalent number of turns of magnet wire.

If the mesh does have enameled wire, and you can make the connections such that each wire contributes to a loop, and you can bend the mesh such that it occupies little more space than the equivalent number of turns of magnet wire, I see no reason why it would function any more poorly than the magnet wire version.  That said, it's an awful lot of work for little-to-no gain.  Get some magnet wire, find a university or business with a coil winding machine (or buy one/make one with an electric drill and footswitch), and do it right.