how long do they set before fish invade set 14 out last week

Depends on location, depth, etc. Some start producing immediately, some never do. It's not the easiest thing to predict or do, otherwise everyone would be a Crappie Guide making kazillions of dollars. Your on the right track though. Larry King on Sam Rayburn had a brush pile set so that no matter the time of year, water temp, wind direction, etc, clients could catch fish. That also includes enough sets for the clients that come back a week later and fish the same spot.

What's the best rope for tying weights to brush that will not rot in water?

Nylon rope will out last the brush.

Good nylon will outlast the device. If you can afford wire, go for it. Recommend you put legs on bottom or jug on center pole. That will keep it upright. At some point someone is going to find it. Good possibly it will be pulled over by fishing line or an anchor.

In my opinion, location is more important than what you use to build structure. I've caught crappie on a single stick with nothing else around it. They want cover and just about anything seems to work. But, I also think bigger is better to attract larger numbers.
Ernest Paty's pvc design has worked well for me and this spring I placed a large magnolia tree and it is producing better than anything else I've built. Don't know how long magnolia will last but, it has the best branching of anything.