It should be strong enough. Copperweld is some strong stuff. I would have no doubts about it being large enough for an 80 meter antenna. Never having had a 160 meter antenna made from Copperweld, I can't say about that for sure, though.
Depending on how the thing is stung up in trees, a pulley and weights, or a strong spring on each end would be a good idea. That should take care of any tree movement. Couldn't say how to measure the strength of a spring, but can tell you about those weights. Hang enough weight on the end of the rope so that the antenna hangs pretty 'flat'. Maybe a three or four foot 'sag' in the middle, something close to that. The pulleys used ought to be sort of large too, around 4 inches in diameter, or so. Smaller pulleys tend to not 'spin' fast enough and can bind, which is a very good way to snap the wire. Regular maintenance is a very good idea with them too (grease).
- 'Doc