There may be an interrelation with damages to the port (in MacBook/Pro)
and the inability for the MagSafe connector to power the computer.
A cure may be to inquire about the cost of replacement, of the MagSafe DC-in Board.
This may be more complex or may require a bit more technical effort for casual users
to attempt a repair replacement of this part; or to diagnose then get the correct part.
This part is referenced in the iFixit.com repair guides for macbook/pro, online.
https://www.ifixit.com/Device/MacBook_Pro_15%22_Unibody - 15-inch 2009
The cure would appear to be, replace the MagSafe port assembly in the computer.
You could get free advice or diagnostic help from an Apple Store with Genius staff
with an appointment; however the unit is probably too old for them to offer repairs.
In any event...
Good luck & happy computing!
You likely have a short somewhere so the charger and/or battery power overheated the connection.
At this point you'll need to replace the MagSafe port and I would strongly recommend you replace the MagSafe charger as well. Here's the IFIXIT guide you'll need to follow: MacBook Pro 15" Unibody Early 2011 MagSafe DC-In Board Replacement. As you can see this is not an easy repair. It maybe best to bring your system in to an Apple Store or Apple authorized service center.
The coating is just cosmetic and surface protection for the magnets (and they do rust since made of iron).
The insulation (plastic) around the pins is important and it still looks ok to me.
There is no fire or short danger, but you might want to re-coat the magnets surface to prevent corrosion, since you are obviously in some high air pollution (corrosion) area.