I can't add to cart without HTTPS enabled
The problem was that commerce_cart.module requires a valid session or logged-in user.  The security settings of the OP's site prevented sessions being sent out to HTTP requests.
While the OP solved this problem via PHP.ini, there are other valid solutions:


Adding session.cookie_secure = 0 in settings.php
Requiring all users to use HTTPS.
Preventing the cart from displaying to anonymous users.