To trace a wire that is cut or has no end, can anyone recommend a tester that works well?

Try a tool made by Triplett called a "Fox and Hound" #3245K. It injects a tone onto the wire and you use a probe to find that wire within a bundle at the other end.

I've got an Amprobe AT-2004 and wouldn't be without it. It's not cheap, but it will find a wire anywhere, in any condition. Here's a link to look at the specs. http://www.amprobe.com/cgi-bin/pdc/viewprod.cgi?pid=269&tid=1&type=elec

I got mine on ebay for about half of retail.

This is the same thing as the Fox and Hound more or less http://www.tempo.textron.com/products/prod.701k.html or you can always try http://www.tempo.textron.com/products/prod.801k.html which takes out the 60 hertz interference of live nearby circuits.