|"Do I need to get a new CPU?"|
"does the warranty cover the damage?"
"Does a CPU still work even if a single pin fallls off ?"
Only if a pin was an "N/C" (not connected) internally. While some of the very old ones may have been, real-estate on a CPU die is so expensive now that it's unlikely any of the pins on a newer CPU is unused...
"Channeling the spirit of jboy..."