Carl's Dad appears in Cybernetic Ghost of Christmas Past from the Future. Carl bares a striking resemblance to his father. In Cybernetic Ghost of Christmas Past from the Future, Carl's father's personality is less than caring.
For a Christmas present he gives Carl a patch of carpet (which he then states is "berber"), as Carl is playing and exclaiming to his father that the patch is a "flying magic carpet" to which he tells Carl that the carpet is going to be dinner, upon Carl's objection to eating carpet, he explains that you have to boil it to soften the glue). He then then tells him to hurry up so they can get to work revealing that Carl started working at a young age. Through Carl's objection, it is revealed that the job in question is making Insulation (it is not stated what type or for what purpose, but the respirator suggests fiberglass insulation for homes).
His status is unknown. The Cybernetic Ghost of Christmas Past from the Future shows Carl his Christmas of 1968, during which his father refers to getting a work permit for an 8 year old, which establishes Carl's age, but not the age of his father.
The fact that Carl wishes for a "new mommy" in 1968 and she is later seen in a Veteran's Hospital suggests that his father served in the military at some point (since only military members and their families are entitled to use Veteran's Hospitals) and they divorced at some point before Carl was 8, with Carl's father taking custody of him and raising him.