"Does it lose any of its stickiness if re-wrapped?"
No, it's perfectly good. I keep several credit cards wrapped with it in my cars (including my work vehicle), and it's just like it came off the original roll. Since all my duct tape seems to collect dog hair for some reason
, before I put it into a tool kit or something, I wrap it with cooking parchment paper*, write "Duct Tape" on the outside with a felt-tip marker and wrap a rubber band around it.
*Cooking parchment is like paper teflon -- not much will stick to it, and you can use it for tinder, too.
Sue