Can cardamom pods really be used whole?

Absolutely. I use them in Indian pilafs all the time. Just pull them out (as you would whole bay leaves, cinnamon sticks or cloves) or warn your guests to watch out for them.

In baked applications (cookies, cakes, etc) I'd use the ground cardamom instead.

Thanks for these extremely helpful answers, everyone!

You can for sure, and you can even leave them in the dish if well cooked. I use them with Daal and they provide some nice flavor when eaten whole.

You can definitely use cardamom pods whole. You'll see this in a lot of Indian or Middle Eastern recipes, especially with pilaf-type dishes. The idea is to lightly scent the dish without giving it an intense cardamom flavor. (Compare it to whole cinnamon sticks or cloves.) You're not intended to eat them whole, although I think some people have a taste for it.

I think whole cardamom pods work better in palak paneer especially because that dish is not meant to be sweet. Rather, it should be spicy hot and full of the spinach and paneer flavors, with a deep undercurrent of many spices (of which the cardamom is only one).

Yup. You either pay attention to what your eating or take them out before you serve the dish as they aren't meant to be eaten, just to spice the dish.