Any way to remove old wiring from wall that is stapled without removing plaster/lath?

Send a cable or wire down the wall and attach it to the bottom of the wire if possible. It should pull up, releasing the staples as it goes. You don't have to remove it, you could just clip it and use a different one.

What is reason for removing old wire inside wall?

If u have access to both ends, u can cut it close to hole and push the balance inside hole.

What type of wire is it?

Chuck

I don't have a new answer ( I usually cut it and stick in wall cavity, like Chuck mentioned) but have a related question: Once you've done this and run new romex, what do you do about stapling it within 8" of the box? Do you just leave it flopping in the walls? I know you can cut holes just above the box and patch afterwards, but that defeats the purpose...

Fished cables do not need to be secured or stapled.

Stan's right. Fished wires don't need to be stapled.

Chuck