Absolutely not. Supplements (even if they have "natural" names like gaia) are minimally regulated by the fda. Their quality can be inconsistent, the manufacturing isn't standardized, and in no way should be given to a 3 week old whose immune system is just developing. Your baby does not need this type of bacteria in it's gut. If you think your baby has stomach issues, please talk with their doctor first.
Not sure. Probiotic drops is still being investigated as to their usefulness in infants. I would discuss this with your pediatrician before adding anything to your baby's food.