I posted a little above on this.
You ask several questions there, I don't know if I have easy answers for all of them, but I'll attempt to answer them as I can.
As I mentioned above, I don't recall any fairytales where they presented Goblins (or Ogres or Trolls) with Children off the top of my head. There may be a story or two of the Goblin King (I seem to recall one or two may have) having a son or child, but i'd have to go through quite a big of books to find which story it was (The most plentiful source of stories I'd say were the Fairy Books by Andrew Lang, these were normally color coded as things such as the Red Fairy Book, the Blue Fairy Book, the Green Fairy book [and my personal favorite], and many more). Grimms Fairy Tales are normally a little bit more simplistic, but Goblins are generally more from English folklore rather than Germanic or other European stories and legends.
Interestingly enough, it was more the Elves and Fairies themselves which may have had children. Elves were NOT like D&D elves generally, they ranged in size and shape and demeanor. Sometimes they were good and sometimes they were very evil. They were far more likely to be married or have children than other creatures of lore. Goblins could be seen as evil elves (for example, Hobgoblins are more of a non-english goblin in origin, but it's basically a term for a Dark Elf...but NOT how we understand Dark Elves to be in D&D).
Normally, as Goblins were the monsters, anything outside of that wasn't really touched upon. They generally were there to torment or hurt the protagonist of the story. Though generally sentient to talk, torment, and threaten, they didn't have children in Fairytales and Lore as a general rule. It's like asking if a Vampire has Children after it becomes a Vampire. Monsters having children is more of a more recent evolution of ideas.
Which brings us to Tolkien. I suppose the first instance of Goblins in caves would be in The Hobbit. There wasn't really much about any children or anything at all about families in that book with Goblins. Goblins were more of creatures that threatened the protagonists of the story and then rode out to battle against the armies of Men, Elves, and Dwarves.