Something from Jenson's site.
http://www.jensonusa.com/store/imgWrapper.aspx?img=/rd/RD603B00_triple.jpg&type=3
So the part that the black barrel adjuster screws into is what is stripped out? More than likely, it probably won't be able to be re-threaded without losing something. In essence, when something is "stripped out" as you are observing, that would mean the threads are damaged in some way - either the barrel adjuster or the part it's going into. Did you completely remove the barrel adjuster to inspect the threads, both on the barrel adjuster and the part it screws into to see what part is damaged? My guess is you perhaps did that?
"Is it possible to rethread it or do I need to replace the entire derailleur?"
Again much of this involves inspecting it. If the threads aren't in good shape on the derailleur part (i.e. won't be fine if you just clean them up and lube them a bit), then you might be able to rethread it to be a little larger and use a slightly larger barrel adjuster (if you can locate one). Otherwise if it turns out to be a show-stopper and the barrel adjuster still won't thread no matter what you do to thread it in, then it probably needs replaced.
"If replacement is necessary, do I need to take the master chain link apart to rethread the chain through the derailleur?"
From what I can tell of the derailleurs I've seen, they usually aren't meant to be taken apart. The only replacement parts I've ever found for them are the cogs themselves and the mounting hardware. So likely the answer to this is yes.
"Is taking the derailleur cage apart an option if I don't have a chain tool?"
Likely no.
But then again, I could be wrong.
Why is it that they make adult bikes that'll generally work for 5'9" or above, yet when you pedal these same bikes they only work for someone who is 5'4" or so?