I've been unable to to determine how the "my cart" function actually works. Sometimes it saves the items I added, sometimes it doesn't. Is there a staple method I can use to permanently store the items I want, so that I can just buy all in one swoop when my pay comes in?

The only thing I can suggest is adding them to the wishlist, but I have no idea if they added a "buy all" button yet =/

I see odd issue with the various promos on cart-save/not_save, but I've understood the cart to be a short term ability, not one that is used to store stuff for days.
edit: as bansama already suggests, wishlist them. There is no option to add everything on the list though, you can buy titles individually, or go to each page and add to the cart then.

I just click the "add to cart" link on the game card (see image), when ready to purchase, click download now and all my cart additions are there. Might work for some and not others.
Edit: The only time I use the cart is when I am ready to purchase, I go to each game, add it to the cart and push download now on the last one.I have never tried to carry my cart additions over to another session so it may only work for the current session. I have all cookies turned off.

What's in your cart wouldn't be saved on GOG's servers at that point, but in your cookies. Chances are you're clearing out your cookies and that's effectively emptying your cart.
So either don't delete cookies while you're still putting together an order, or as stated you could try wish listing them for easy reference.