There are no lanyard holes on the Zip. But you could loop a lanyard strap (without using a clip on it) through the Zip’s clip and hang it that way.
You also could purchase something like a Tek Tab (very low-profile) to glue onto the clip, which has a metal loop on it for exactly the purpose you want. http://forums.sandisk.com/t5/Sansa-Clip-Zip/Nearly-free-substitute-Clip-clip-after-rebate/m-p/276986/highlight/true#M3081 Note: these still are available free-after-rebate! Although there currently is a $1.99 shipping charge for them (for awhile, shipping was free), if there isn’t a local store near you. This may be exactly what you could use, and even comes with its own lanyard and carabiners.
As to screensavers, the Clips can be set to automatically turn the screen off after a certain number of seconds–while the players can also be set to have the screen always on, best to let it go dark, to save power as well as preserve the screen (which can wear out, over time).