Fried Pins- Output to Output short

Device: Arduino Uno
Chip: ATMEGA328P-PU
I accidentally used terminal 13 as my ground terminal.
It worked for some time, and I did not notice the error.
Now I noticed that there are some problems with digital terminal 13.
I am unable to upload a sketch to the device now if there is something connected to 13.
And sometimes it provides a voltage even when output is not enabled on this terminal.
Other than that, If I stop using this pin everything else seems to work fine.

If I replace the Atmega chip will that solve this problem? or is the damage on the board?
Is this the correct replacement chip for the Arduino Uno?

Grounding an output pin will damage it or other parts of the chip nearby.

Yes, replacing will solve.
No, board is not likely damaged. Just the chip.
Yes, that is the correct part.

Really you can keep using that one, unless you need to use pin13 for SPI or some such thing.
Nothing wrong with replacing it of course.
The chip's internal interconnects will almost always pop before the board overheats and fries, your board should be just fine like CrossRoads said.

While you CAN keep using that chip, it would probably be a good practice to replace it (and throw that one away.) At the very least, if you need to use it, put a sticker on the chip that says "bad 13."

Down the road, you will forget about this problem and end up troubleshooting something you already knew about.