Are babies ever tested for std's during any one of their early check ups or at birth (even if mother isn't suspect of anything)?

Depends. Most pregnancies that receive prenatal care will have some basic screening done during the pregnancy. If someone without prenatal care comes in for a delivery, some obtain some basic tests, like that for aids & syphilis & may look for other issues if any symptoms are evident. After birth, infants do not receive routine teats for any std's.