Can local anesthetic hurt?

It does. Local anesthetics set off the nerves they are putting to sleep. There is an initial sensation as the nerve discharges, but then it can't recharge and remains asleep. Injection pain can be reduced with slow injection and buffering the ph of the anesthetic.
It can sting. A little. If injected slowly, through a small needle, under low pressure - it's much better. Fortunately, it works quickly.
Yes, but less than. Surgery without anesthesia! it's just a tiny needle stick and a bit of a burning sensation as the local anesthetic works--and it works almost instantly! it's truly the lesser of two evils; even a mole removal would be too uncomfortable without local anesthesia, so buck up and do not fear theneedle of local; it's really better and easier than you worry about!