Electrical Outlet arc burned the wire. Do I need to replace the entire line?

The rest of the NM cable should be fine. When I install outlets like this, I pigtail the wires and connect the single wires to the outlet and still use a shepherd's hook behind the clamp.

The closeness isn't a problem if done right. You can have a split outlet with different legs so you've got 240 volts just that close.

Is this GFCI necessary?

First, before installing a GFCI outlet, check for the outlet being GFCI protected elsewhere. It could be a GFCI breaker, or a deadfront recep (no holes for plugs), or another GFCI outlet in the kitchen. For older installations it could be a GFCI outlet anywhere, since kitchen counter circuits were once allowed to serve other loads.

The fastest way to do this is install a plain recep and use a GFCI tester, assuming the ground wires are good throughout.

Any outlet that is GFCI protected needs a sticker saying that. Or an artsy faceplate or whatever, they're not particular.

GFCI devices are able to protect downline loads. That is the solitary purpose of the LOAD terminals, which should never be used for any other purpose. It's easy for Johnny Sophomoric (who learned just enough to be dangerous) to make a whirlwind tour and slap GFCI receps everywhere and create a fiasco where resetting it is like playing Myst.

Repairing the wires

Yes, just strip the insulation beyond the obviously-damaged areas, and cut the copper back to beyond where it is discolored (annealed).

At this point you must have 3" of wire beyond the face of the wall, and 6" beyond the cable clamp. If you don't have that, you'll need to pull more wire through from the wall, or replace the cable.

This is why I discourage cutting excess wire length.

As far as the device's near connections, I would check the device for a UL, CSA or ETL listing with a file number. If that is present, then it conforms with UL White Book safety standards. If it says "CE" or not much at all, it may be cheap Cheese junk direct-imported (including bought off the scummy Amazon Marketplace)... search the house for any more like it.