No, the child doesn't get stoned. In fact, the mother doesn't get stoned until the child is weaned.
Depends on the sect. There are different Sharia requirements for each of the various and sundry sects, plus, the "court" metes out the sentence based on their "interpretation" of the Quran. Just because it may run against your belief system, don't think that these animals are above stoning a baby. I have lived and worked among them, You can not even begin to imagine the depths of depravity that a Sharia can foment.
Semper Fi