Does a baby remain safe even if the expecting mother does not eat much?

Generally yes. Babies tend to get what they need at mom's expense. If you are underweight, there is a higher risk for growth problems if you are not consuming enough calories. In a normal pregnancy, you only need an extra 300 calories per day in the third trimester from what you would normally eat when you are not pregnant. In spite of popular belief, you are not really "eating for two".
Depends on time. In early pregnancy..Especially the first trimester, many women don't gain much weight..Some actually may lose some because of n/v due to morning sickness etc...Tiny baby does not need much food at this stage, but will need more and more later on. Recommended wt gain is around 25-30bs for the pregnancy. Follo-up regularly and u will be advised accordingly with your visits. Congrats and good luck.
Fetal growth. During the winter of 1944 the dutch population experienced a near starvation diet. The total calories per day were significantly lower than a normal diet. The women pregnant at that time still had healthy babies, but the babies' weights were somewhat lower than usual. These babies were followed and developed normally.