WD40 is cheap and penetrates pretty good, I like the PBblaster for bolt threads better. WD is a good cleaner too. That uninsulated wire is a ground for the socket base to the lamp housing, you can re solder it.
There is a cross head screw to remove the socket for the housing. I can see it through the glass, it is below the bulb. Not much slack in the black positive wire there, unless you can pull out the black rubber seal plug and undo the black from the harness.
A lot of work for just a turn signal lamp, huh? Hopefully your socket can just be cleaned up and the spring in the base is still working to keep the bulb contact contacting.