producing ozone through electrolysis of water
First of all $\ce{H2SO4}$ produces $\ce{H2SO5}$ which is a peroxyacid during electrolysis  which most electrolytes will not do, so I suspect that $\ce{H2SO4}$ will be mandatory unless phosphoric acid ($\ce{H3PO4}$) can.  This peroxyacid give you a potential for a mechanism of ozone creation.
$$\ce{H2SO5 + O2 -&gt; H2SO4 + O3}$$
Note: I do not claim this to be the mechanism but only an explaination for why the $\ce{H2SO4}$ may be important.

The carbon and platinum electrodes are not only chemically resistant, but may likely have a catalytic effect in producing ozone.  You can get platinum wire at a reasonable price if you look around.

In regard to testing if you have ozone, you can use paper strips that have potassium iodide impregnated in them.  The ozone will oxidize the iodide ion and turn the paper strip brown.