A puppy with no anti-rabies shot, bit me in my thumb just below the nail. Is that enough to give me rabies?

Could be! It would depend on where the puppy had been and what it had been exposed to. If it had come in contact with a rabid animal or carcass, or other rabid animal, and licked, chewed or faught with it, then you could get the rabies virus infection. You should quarantine the animal for 2 weeks, or have animal control do it and notify them immediatly and seek advise on rabies shots. Rabies is deadly!
No. Puppies would be extraordinarily unlikely to be carriers of rabies. Puppies bite people and children all of the time as a normal exercise of their practice chewing. You shouldn't worry about this at all.