Not really familiar with lead free solder, but according to this site it shouldn't make a huge difference when doing general purpose soldering. As long as it has enough tin (I'd say at least 70%) it should be alright.

I've also never tried the copper wire trick you're describing, but it shouldn't melt or damage your iron if it's just wrapped. Solid core wire (as opposed to stranded) would probably be what you wanna look for, and gauge wise I would think anything finer than the tip of your iron would be okay. Might want to wait and see if anyone else here has tried it before though, just in case there's something I'm missing here.