Their first child was born with severe birth defects and did not live past birth. At least this is what I have read. I don't recall that the babe was even given a name.
What kind of impact would this have on a man who believed in folk magic and spiritual voodoo? I would think it would freak him out, but it didn't seem to phase him.
The death of Alvin seemed to be the biggest lose in his life.
Maybe Alvin represented a father figure to him.
I just find it odd there isn't more said about the firstborn (?son).
Does anyone have any detailed reference to this baby?

They also had twins that died. Not to worry. They were replaced with someone else's babies. Apparently the mother had died, so the twins went to Joseph and Emma. I don't recall how or why their twins died.

Perhaps old Joe was the father of these kids as well??? just saying...

Hmmm. That never occurred to me. It makes one wonder.

I understand that Joseph had made predictions or revelations that the child would translate the Gold Plates, or be part of the process somehow. The child was supposed to be gifted but turned out deformed and stillborn.

The information should be on one of Uncle Dale's sites IIRC, if anyone is up to searching, it has been a few years since I read about this.

http://solomonspalding.com/

Here is is, witness statements from Howe's Book:

"Joshua M'Kune states, that he "was acquainted with Joseph Smith Jr. and Martin Harris, during their residence in Harmony, Pa., and knew them to be artful seducers;" -- That they informed him that "Smith had found a sword, breast-plate, and a pair of spectacles, at the time he found the gold plates" -- that these were to be shewn to all the world as evidence of the truth of what was contained in those plates," and that "he (M'Kune) and others should see them at a specified time." He also states that "the time for the exhibition of the Plates, &c. has gone by, and he has not seen them." "Joseph Smith, Jr. told him that (Smith's) first-born child was to translate the characters, and hieroglyphics, upon the Plates into our language at the age of three years; but this child was not permitted to live"

and,

"SOPHIA LEWIS, certifies ... She states that she heard Smith say "the Book of Plates could not be opened under penalty of death by any other person but his (Smith's) first-born, which was to be a male." She says she "was present at the birth of this child, and that it was still-born and very much deformed.""

http://www.solomonspalding.com/docs/1834howf.htm

This is a new one to me. I've never heard that before. It sounds like JS was going to give himself 3 years or so to put his book together. Interesting.

I don't know which is harder to believe, that he translated the plates, or that a three year old was going to do it. That would have made for interesting twist in the story. People would have felt bad picking on a little 3 yo child. It just goes to show that he would use anyone to make money and further his scams, even his own toddler,

I've often wondered if Rhesus factor incompatibility might have caused their reproductive problems. I have Rh negative blood, and my twins inherited the Rh factor from my husband, so I had to have shots during pregnancy and after birth in order to block my system from making antibodies against the twins or any future fetus, which I never plan on having, but I took the shot in case of accident.

The damage the antibodies do to a fetus is devastating and mostly ends in miscarriage. Those who do carry to term usually don't live long postpartum and can be deformed.

I've also wondered if JS wasn't only looking for variety with polygamy but also healthy offspring.

Just to answer the question about what happened to the adopted twins. As I understnd it, the boy died in childhood, after beig exposed to he cold during his fahter's tarring and feathering. He had measles or something at the time. There is now some controversy over why JS was tarred an feathered them, possibly related to polygamy.
The boy's twin sister did not go west with the Church, but stayed with Emma. She had a bad marriage, and returned to live with Emma. She died young of breast cancer, after being cared for by Emma.

Joseph's mother briefly mentions the birth in her autobiography, but does not give a name.
Joseph F. Smith edited the manuscript in 1901. He adds the date of birth, 15 June 1828 and place of burial as "the old cemetary in Harmony" but does not add a name.
In her listing of family members she does not list the firstborn.
Oddly she lists the adopted girl but not the adopted boy.