Lead is used sometimes in rubber/plastic to make it more flexible, but there are other ways to accomplish the same thing so to answer your first question: No. (See 1st source.) Lead solder is pretty common in the US, although it is becoming less so. However, an electrical cord doesn't generally have any solder in it, solder is used for low power items, not for extension cords.
The label is required if you sell your product in California. It doesn't matter where it is made; if it is being sold in California and it has lead in it, it has to have the label. (See 2nd source.) Some companies use the same labels in all states, some products ship the same thing to California with a label on it, and to the other 49 states with no label. As such, the only way I know of to buy an extension cord that definitely doesn't have lead in it, is to buy it in California and make sure it doesn't have a label on it. (Keep in mind that Prop 65 covers more than just lead, but for cords lead is the most common thing that they'd have in them.)
I do care. While Tom is right that I'm smart enough to not eat my extension cords, I hate to have to wash my hands after handling one. And they do fray/break down/etc and then the dust becomes airborne and I breath it. And my cat sometimes chews on things, (and I don't want her to die either,) as do dogs/babies/etc. However, California is a bigger state than mine, and Prop 65 is not without its problems so we haven't passed one like it.
Europe has taken a different approach to this problem and has all but flat out banned lead with exceptions for things like car batteries and X-ray shields. (See 3rd source.) This too is not without its problems, but as time has gone on the market has adjusted. The interesting side effect of this is that companies that sell internationally, (mine does,) have taken all the lead solder out of our assembly lines because it just isn't worth the trouble of having one product for Europe and one for the US. It is that way for many companies which means that most new computers in the US don't have any lead in them at all. (I still wouldn't eat them, it seems like it would be hard on your teeth.) Of course, the non-lead solder has gotten better over time to, so even the companies that don't have a reason to switch are switching anyways because it is better and doesn't cost anything to do it.
Unfortunately, for extension cords with NEMA [US] blades/sockets, they are obviously not an export product, so they might continue to have lead based plastic/rubber in the insulation.