That is quite bizarre and I have absolutely NO IDEA what it is! Maybe a tiny alien spacecraft in disguise?

At Home Depot today, in the indoor-plants section, I've bought a pretty, tropical-looking plant (the only one there of its kind) that the label had fallen off of. The Home Depot clerks could not identify this plant, so they sold it to me at their cheapest price for plants in ten-inch pots (the size of pot it came in). In hopes that you can tell me this plant's common and/or scientific names, and direct me to care-instructions, below I'll describe the plant.
This plant (half-dead and loosely potted in loose soil when I got it) has stems 8 - 12 inches high: each stem from one seed clearly visible in the loose soil without my disturbing them to see them. These big, round seeds measure about the size of a very large gumball with fibrous roots coming out. (Think of an avocado pit, only smaller and not "split" the way that avocado pits sometimes split when they sprout.) The stems themselves each have a fleshy, rounded-cone-shaped base that tapers upward into the actual stem. On each stem, starting about halfway up, leaves grow in pairs with about an inch or two between each pair and the next: above the topmost pair on each stem grows a single leaf. The leaves look very much like rhododendron leaves - 2 - 4 inches long, and about 3/4 of an inch to 1 inch wide - and have a texture in-between that of rhododendron leaves and avocado leaves. What have I bought, and what do I do to take care of it properly (indoors, of course)?

The tapering stems and leaf shape, arrangement, and texture you describe sound like a malabar chestnut (bombax) but the seed of this tree is not round, it is similar in shape to American Chestnut.