If I faint, will the baby get hurt?

Depends. Depends on the reason you fainted, how long you're unconscious, and whether you had trauma to your abdomen. . Also depends how far along you are as in early pregnancy, trauma is not likely to be an issue.
Not usually. The only significant danger is falling. If you fell on an object across the abdomen, the fetus could be at risk. Usually a baby is very well protected and cushioned in the womb and is usually safe. Once a patient faints, usually from a temporary drop in blood pressure, the pressure recovers quickly without harm to the baby or mother. Driving, of course could pose different risks.