Water behind the walls can cause all kinds of problems. Rot and mold may exist behind that pretty pink, and it may not be able to be salvaged. What is on the other side of the shower? Is there an inconspicuous spot that you could cut a hole and repair it later? That would allow you to view in behind the tile without ripping it out to do so and allow you to see if there are any problems.

One of the walls of my master bedroom is on the other side. There is a heat/AC air return vent in the wall that I could probably remove but I don't know how much I would see. The water that is gotten through the wall just drips down onto my basement floor--as in the ceiling in that part of my basement is not finished. I can see all my plumbing pipes and the bottom of my tub.

Based on the age of the home, what materials were most likely used behind the tile?

You can about bet it's a mud job. Alot of times, if they didn't soak the tiles long enough before setting them, over time, they'll lose their bond to the wall, and then you get the situation you have now. So long as that mud is in good shape, I'd leave it there, strip the tile, waterproof the mud, and reset the tile on the original mud bed. One way or the other, from what you've told us, I'd be willing to bet that tile's is ALL ready to come down.

We don't have the $$ to take down all of the tile right now. Is there a temporary quick-fix that I can do that will last a year or two? Also, we plan on keeping our retro pink bathroom. :) It is a lovely bathroom.

Highly doubtful. Of course it may not cost you anything to take it down-- it may come down all on its own. You might want to take a butter knife and tap all the tile. If you get a hollow sound from it, get ready.

Hmmm, rats!
This is the tile I pried off with a screwdriver this morning. First tile row above the tub. You can see where the grout is missing.

This is what most of the tile in the shower looks like, this is near the top of the shower.

And some examples of the random missing grout.

(Sorry for all of the photos, and thanks for all of the advice!)