Can a woman with syphilis give birth to a healthy new baby, as long as she got antibiotic treatment?

Yes. As long as pregnant mother gets proper treatment and follow-ups.
Yes and no. As long as mom got adequately treated it is possible that she will have an uncomplicated pregnancy and baby will be fine. However, if the infection had a chance to affect baby before treatment then treatment will cure mom but not necessarily reverse the injury to baby. This is why early prenatal care is critical. A syphilis test is a part of early care so this kind of situation can be avoided.