Problems:

1. Click on cart or checkout or my-account and 404 not found
2. Clicking shopping cart red item delete button doesn't delete the cart item.
3. Cart only worked with permalinks set to "Default" i.e. not any rewriting

Possible cause:

Bad nginx configuration

# before
location  / {
        try_files $uri $uri/ /index.php;
  }

# fixed 
location  / {
        try_files $uri $uri/ /index.php?$args;
  }