No, a lot of PVC wire insulation contains no lead. For example, all wire meant to be sold in the EU must be ROHS complaint, and thus contain no lead. Many US manufacturers are also compliant, and the number is growing. Many manufacturers state this explicitly on their websites (e.g. Tripp Lite and Southwire, to take two random examples). Many others can be identified by either an RoHS designation or lack of CA Prop 65 warning for lead.
The answer to your second question is yes, handling cords with heavily leaded insulation can cause significant amounts of lead to rub off on your hands. This is easily seen with a Lead-Check swab, which will turn red when wiped either on the cord, or even on your fingers after you've spent some time handling the cord.
> all wire meant to be sold in the EU must be ROHS complaint
Just an aside, this is a personal gripe. I grew up on lead solder, making Heathkit projects with my dad.
I tried re-engaging with the electronics hobby as an adult, decades later, in a world filled with ROHS solder, and I have to say it is a huge pain in the ass to work with. Adhesion, balling issues, etc, regardless of core type, flux, etc, even on nice controlled-temp Hakko stations, versus the Weller crap irons of my youth.
Maybe I'm just a grumpy old man, or maybe my memories are covered with rose glasses. But electronics seemed be more enjoyable in the lead era. And to my knowledge, my IQ did not suffer.
I suspect the opposite. I grew up eating off of lead-contaminated Lenox dishes, playing with fishing weights and mercury from broken thermostats, and other stimulating but toxic activities. I strongly suspect these things account for at least some of the attention deficit and impulse control issues I now struggle with as an adult.
I'm sure your IQ is more than adequate, but you have no way of knowing what it would be if your brain was kept cleaner in your youth. For my part, I'd love to be smarter.
You can't get lead poisoning from using lead solder. The melting point is not high enough to turn the lead into a gas. In fact, lead free solders are generally more harmful - due to higher melting points they tend to produce a lot more flux fumes. ROHS solder was introduced so when you junk your mass produced electrical appliance, you're not adding more lead to the environment. For hobbyists it makes no sense.
The answer to your second question is yes, handling cords with heavily leaded insulation can cause significant amounts of lead to rub off on your hands. This is easily seen with a Lead-Check swab, which will turn red when wiped either on the cord, or even on your fingers after you've spent some time handling the cord.