I was sold a car with the entire headlight system dependent upon an old extension cord. Recourse?

No. Really you know you shouldn't use any cord, but that cord uses wire type 16/3 SPT-2, which according to National Electrical Code Section 400 for cords says is rated 13 amps, "damp location, not hard use". http://freenec.com/T220.html http://freenec.com/T224.html Your circuit breaker feeding it will be either a 15 or 20 amp breaker.

That cord could overheat without tripping the breaker, and if used long term you could begin to forget the cord is there and subject it to "hard use" damage. I would use a #14 awg (14/3) cable (smaller number is bigger wire), and a wire type with a "J" in the cable type for "hard use" or H for "extra hard use". This cord would be better http://amzn.to/1OpfJyh.

If the extension/power bar is rated the same as the outlet, then you can run lots of low current devices (TV, games console, computer, radio, phone charger, bedside light etc. quite safely. Just don't try to use a heater or a kettle, as these take a large current. Even if you do overload the extension, the fuse/breaker controlling your outlet should protect you against any damage/danger.

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