It sounds like the short caused the receptacle to melt and then tripped the breaker for the circuit. That would explain why you have no power to anything else despite everything being wired on the "Line" side of the receptacle instead of part on "Load".
Check your breaker panel for anything tripped. Keep in mind that with many breakers it is not really obvious when it has tripped. If you can't find anything tripped (and if you are not sure, just turn a breaker all the way Off and then back On), post a picture of the panel.
As far as why the receptacle melted rather than simply "tripped", a short in the cord would not be a ground fault (which would trip extremely quickly) but rather be a normal over-current situation. As long as hot == neutral, the GFCI won't trip and you are relying on the circuit breaker in the panel to protect you. That circuit breaker might be relatively slow to trip, depending on many factors. At the same time, the GFCI is going to be overly sensitive to an overcurrent situation because of the sensitive electronics included in it. It shouldn't melt, but that is not so surprising.
As far as the black markings on a white & ground wire, I suspect that was someone's way of trying to distinguish between the "in" and "out" sets of wires. The matching black wire might be marked with a black marker too, but that wouldn't be so obvious :-)
Based on additional information in comments, it appears that there are some receptacles on the circuit that are still working but nothing "after" the melted GFCI is working. Based on the description that the wires are pigtailed to the "Line" side rather than working off of "Load":
- Disconnect the pigtails from the GFCI to eliminate that from having any effect (not sure how it would, but just in case - and that GFCI needs to be replaced anyway).
- Check the last working receptacle. You may find that the wires heading to the melted GFCI show signs of melting/arcing/burning. If those wires are backstab connections then moving to screws may help. If not, try replacing that receptacle and see if that resolves the problem.