OK. I got the scoop. I think I figured out what happened here and why it looked the way it did. But first, here's a few pictures of what I've been trying to describe:
Here's a picture of the flange that I've been talking about
Here's the space it's situated in:
Here's the marble slab (now broken apart into 2 pieces):
What happened is I took a crowbar to try to raise the marble slab in order to check out how I could disengage the flange. The marble slab then broke into two pieces. Although I was sorry to see the marble slab broken, I was then able to check out the flange more closely. I'm not sure what the flange is made of, but it's a hard metal, and it's definitely soldered down solid (the house was built in 1965). The flange, coming from the floor, is kind of dog-legged. I'm sure you professionals have plumbing terms for these things, but since I don't know the correct terms, I'll just describe them the best I can. From what I can guess, the plumbing stuff underneath the floor was put in too close to one of the walls. So, in order to center the toilet better in the space allowed, they used this type of dog-legged flange to move the center of the toilet over 4 inches from the opening in the floor. And because they had to use this type of raised flange, they had to use this marble slab in order to hide the raised portion.
So what do I do now? Here's my guess. I'll go ahead and ceramic tile the floor. It looks like I have to keep the flange as is. Then, I'll thin-set the marble pieces back into place and glue them together. I'll then thin-set ceramic tile (since it has a crack in it now) onto the sides and on the top of this marble slab.
Does this sound right? Does anyone have any good suggestions for me?