What is the name of David and Bathsheba's first born who died? Was it Shimea?
From the time the child is first mentioned in II Samuel 11:27 until his death in II Samuel 12:23, the child is not referred to by name. Since II Samuel 12:18 specifically mentions that the death occurred seven days later, if we assume the prophet, Nathan, visited David on the day of the child's birth, then the child was not circumcised, since this happens on the eighth day. It was traditional in the Israelite culture to name a son on the day of his circumcision. See the naming of John in Luke 1:59-63 as an example of this. The naming of Isaac also implies this (Genesis 21:3-4).
The names of David and Bathsheba's sons are given in II Samuel 5:14; I Chronicles 3:5; and 14:4. There were four sons named. Shimea is does appear in I Chronicles 3:5, but he also appears in the other lists under alternative spelling "Shammua" The name means "renowned." In the lists, notice that Solomon is named as the last of the four sons, though we know he was the second child of David and Bathsheba, thus the lists are not in chronological order and nothing further can be drawn from them. It is likely that the first child is not mentioned at all in these lists since the child didn't live beyond a week.