Soon after birth my baby had small pimples all over. Is this common?

Yes. Babies can have neonatal acne for a few weeks after birth, mostly due to maternal hormones exposure. It's benign in most cases.
Yes. This type of rash is commonly known as erythema toxicum, or etox, and is sometimes mistaken for "flea bites." it generally comes and goes for the first few weeks of life. They do not bother your baby, and no treatment is required. If the spots look like blisters, your infant isn't feeding well or appears ill, they need to be examined.