Out of warranty MBP ,Burned Magsafe port and frayed charger cable - repair option and cost?

I have a MBP 15" 2011 model and has is the burned magsafe port (leftmost pin melted) which i noticed today.

The connector gets too hot sometimes ( but remains ok when I charge in sleep mode). Sometimes the charging will stop and LED will become green and then I reconnect it to get it in charging mode.

The Frayed cable: very light cut at connector side

and heavily frayed at brick as shown.

So what should I do now and how much would it cost to repair this? Should I buy a new charger?
Also I live at a place where there is no Apple Service Center (nearest one is 300 km away). So, can I use these things as they are for 2 weeks i.e. by charging in sleep mode or when the port doesn't get hot while charging.

The MagSafe board is a very easy repair for someone who has patience and a small amount of training fixing any computer at all. The MacBook Pro aren't as fidgety to repair as the iMac and MacBook Air (where I would say you should be a skilled technician with official training materials available to work on them reliably).

You're looking to replace this with a handful of screws turned.

Go to iFixit.com and look up the repair guide and parts list for the DC in board and you will know if you have the skills to find the right part, the right tools and the right manual to undertake this work. Otherwise, pay a local technician to do the job/source the parts/give you what they feel is a good guarantee. As for the adapter, buy a new or used one on the aftermarket or from Apple. When the cord is crushed and frayed, it is designed to fail in a way that won't cause a fire hazard. I wouldn't mess with that end of things or with repairing the transformer itself.