The tip of the cart melting is actually common. If you look at the bottom of the cart, you will notice it is rectangular. It is designed to line up with the bridge of the atty. If you put the cart in the atty and don't line these up, the cart will be touching the bridge and ultimately melt. Some people cut about 1/8 of an inch off the cart tips to avoid this.
Yes you can drip directly on the atty. This is called "Direct Dripping". A lot of vapers use this method rather than filling up carts.