Can a lumbar puncture paralyze me?

Rarely. I have seen this happen twice in the span of my medical career, both instances in which the performing physician was not aware of the patients' use of blood thinners. Hematomas formed and extended upward from the "safe" levels of lumbar puncture up to the spinal cord.
Possible but RARE. You would have to have a major complication which is exceedingly rare and possibly related to some other underlying disprder suvh as hemophilia.
Rare. All procedures carry a risk. For that matter, driving your car is one of the riskiest things you do on a daily basis. Procedures, such as lumbar puncture, could cause bleeding or abscess formation, and could lead to paralysis. Usually there is a good reason to do it though.
That. Would be like getting struck by lightening. Also lp's are done below where nerves which could paralyze you run.
Interesting. This is a rumor started in eastern big city ghetto environments by parents to keep their kids behaving. If you are bad, "the doctors in white coats will stick a needle in your back and paralyze you". So far, haven't seen this outcome.