needle size is just too anal for me to wrap my mind around. There are no hard and fast rules except you don't want thread too big to go through the needle eye. As long as you are using standard, readily available thread that should not be a problem. There are charts linking thread size to needle size but frankly I don't have the brain cells to devote to that and I have gotten along reasonably well. If you really want to get scientific, knock yourself out. I use large needles because my eyes can't see the eye on a smallish needle. Makes it tough to thread.
I would use a sharp or universal needle for the application you are suggesting. The idea that tulle would use a ball point makes some intuitive sense. Generally speaking the only fabrics needing ball point needles are close knits and stretch fabrics like high lycra content lingerie/swimsuit and leotard materials. Those are not widely used in hammock gear making. The reason for the ball point is to avoid splitting yarn fibers. A sharp or universal needle will pierce right through a knit yarn resulting in a weakened spot in the fabric. Any stress on that spot will tend to break the fibers and create a hole. In knit fabrics a hole translates to a run. Not a good thing. Talk to a lady friend about runs in hosiery unless you have first hand experience. A ball point needle will not pierce the yarns... but rather push them aside so holes, weak spots and runs do not occur. The same dynamic is a work with the high stretch lycra fabrics. Netting like tulle is not the same kind of situation.
IMO the most versatile stitch on the sewing machine is the straight stitch. You can use the zig zag if you want. But to be honest, the only time I use the zig zag is for garment buttonholes and the occasional reinforcement bar tack. I don't even sew webbing loops with zig zag any more. I prefer the straight stitch for a variety of reasons, not the least of which is most of the commercial webbing I see is done with straight stitches. But that's a personal choice. Either one will hold the things together.
Hope that helps.