For the original scheme, I do think that 1/8th inch will work. I don't really have a good idea about the eyebolts or turnbuckles.
For the new scheme, I'm not at all sure I understand your idea. But I suppose that you are thinking of putting up 3 cables that are parallel to each other but at different heights. The cables would be high enough to not interfere with basketball or volley ball, and so they could remain in place all the time. You'd put in vertical sections of cable, attached by a ring at the top, and to the net at the bottom. Because the 3 horizontal cables are at different heights, the vertical cables would have 3 different lengths.
There might be 3 sets of 10 vertical cables, so the spacing between cables would be around 6 feet.
I haven't given this much thought, but there are two related likely problems.
1) The horizontal cables will sag, and there will be a tendency for the tops of the vertical cables to slide downhill toward the center of the cable span. Maybe there will be enough friction to keep this from happening, but I don't know.
2) The bottoms of the vertical cables will move toward the center of the net, and there will be significant sag in each section of the net between the vertical cables. The roof of the net will also sag. The amount of sag depends on the length of the cables and the weight of the net. I don't know how much. So you would need to space the 3 upper cables further apart than the width of the net, and make the horizontal cables longer than the length of the net in order to keep a reasonable tension on the net itself.
It is possible (but not easy) to estimate the needed configuration of the cables. Frankly I don't want to calculate that unless you're pretty sure that you'll be allowed to go forward with the idea.