Cart always says "0 items" when products are added, and sometimes empties completely on ot

#1) My client is getting the first bug in Firefox although I'm not getting in myself... but one other customer reported it too: Sometimes clicking "Add to Cart" on a product page doesn't seem to do anything. But when they refreshes that same page, the product is in the cart. But they have to refresh for it to show.

#2: In Safari at least, the cart in the header always says "0 item(s)". Always. Whether it's expanded or not (View Cart / Hide Cart).

#2b: Still in Safari at least, although the cart says 0 items are in it, only some pages will show the products that have been added to the cart while others show an EMPTY cart. I've noticed single products pages show them, for example: ....while product category pages and regular pages show an empty cart. Examples: and