I moved into an apartment and I think the toilet was leaking small amounts for a long time. When my husband and I moved in, I caulked around the toilet because that is what I have always learned to do. Now there are air holes through the caulking and every time we sit on the toilet, a weird gray ooze/liquid comes out of the bottom. Is it rotting underneath our toilet? Should it replaced?

You said "apartment" so I assume you are renting. Call maintenance department. You have a leak between the toilet and the drain. That is corrected by replacing the wax ring between the toilet and the drain pipe flange.
Caulking around the toilet prevented the water from running out and now the floor is rotting.