Magsafe port on Macbook Pro Unibody gets extremely hot

Hello!

After a couple of the latest updates, the Magsafe port on the unibody gets extremely hot when the computer is plugged and I'm working on it. Already I can see signs of burnt, melted surfaces both on the head of the cord and the port. Also today I had the SMC reset, because the charger went crazy and it's only charging in some hard to get positions.

When I opened up the back of the computer to disconnect the battery, I could see sticky dirt on the inside of the Magsafe port.

If the Magsafe is connected at the right position, the computer charges correctly only during Sleep mode and when Shut down, if it is working it gets so hot, I can't even touch it.

Besides, the whole computer gets incredibly hot on the bottom, though the fans are not running fast.

All of this occurs when the power is connected! and since the latest updates to Mavericks!!!!!

I attached some photos, where you can see that the two pins on the side are totally burnt.

Is it possible to replace the port and the Magsafe? I'm wondering what happens after I change it, because to me it seems the problem starts with the OS and then deteriorates by itself.

I have a late 2011 Macbook Pro Unibody, running the latest Mavericks OS x 10.9.4, with 2,8 GHz Intel Core i7, 8GB of memory and a Samsung SSD as a startup disk and an 750 GB HDD installed in the Superdrives' place.