Can a CPU melt but still functions in the long run?

115+, C or F??? If Celsius, your CPU is dead. It should have shut down at about 75 C. If it's doing anything at all, it's just limping to it's grave. Replace it.

If Fahrenheit, continue to use it. If it hiccups, burps or acts weird any any other way, replace it.

Next time, take it to a Pro, they're worth every penny!!!

The Machine will still run fine, it did the emergency shutdown to prevent damage to motherboard.

Try and gently remove the stuck cable. If not I suggest replacing your heatsink fan, they're not usualy expensive, about £10-£15. If its a vital cable, not one that has a twin such as a molex connector, then I sugges you could cut it away from the fan, (if you cannot remove it cos the cables melted) and connect the two new ends together with one of those block connectors, or twist them and put some insulation tape on the join. That should solve it and your motherboard should work. For goodness sake make sure heat management is on.

I've seen computers overheat before like that and they ran fine. Its definitely not a good thing to do but it should be fine. The overheating smell is probably the fans motor burning out. Keep an eye on it if you smell it again.

your heat gel is probably fried underneath the heat sink, check that first if it is then there may be permenant damage