The title sums it all, I just am seeking an answer in the Middle Earth world, not the Tolkien nor the Peter Jackson's world. I believe it is already a singularity to find a dwarf attracted to an elf and vice versa. So is it possible for the two breeds to reproduce? If it is, what can the outcome be? A hybrid, and dwarfish elf or an elvish dwarf? Please do excuse my lack of knowledge of the middle earth, I only watched the movies and didn't have the chance to read any written material about it, so if this is a redundant question, just let me know.
We don't know. In Tolkien's writings, no dwarf and Man were ever in a relationship. However, I'm going to go out on a limb and say no, probably not in Tolkien's conception.
There are only two kinds of "interspecies" breeding we ever hear about: Men and Orcs, and Men and Elves.* And in both cases, these are explained by the members being "branches of the same race."
Men and Elves could interbreed because they were both the Children of Illuvatar and were biologically identical:
The existence of Elves: that is of a race of beings closely akin to Men, so closely indeed that they must be regarded as physically (or biologically) simply branches of the same race. (Morgoth's Ring)
The origin of Orcs is debatable, but the most common explanation is that they were in origin either Elves, Men, or both:
Most of them [Orcs] were plainly (and biologically) were corruptions of Elves (and probably later also of Men).
Dwarves, however, were not created by Illuavatar, but by Aule, who had no real clear idea of what the Children of IIlluvatar looked like:
And Aulë made the Dwarves even as they still are, because the forms of the Children who were to come were unclear to his mind, and because the power of Melkor was yet over the Earth; and he wished therefore that they should be strong and unyielding. But fearing that the other Valar might blame his work, he wrought in secret: and he made first the Seven Fathers of the Dwarves in a hall under the mountains in Middle-earth. [...] Since they were to come in the days of the power of Melkor, Aulë made the Dwarves strong to endure. Therefore they are stone-hard, stubborn, fast in friendship and in enmity, and they suffer toil and hunger and hurt of body more hardily than all other speaking peoples; and they live long, far beyond the span of Men, yet not for ever. (Silmarillion, Of Aule and Yavanna)
So we don't know, but it seems unlikely. As an additional datapoint, we can consider the unlikelihood of such a match - the strife of Elves and Dwarves goes back to the very beginning. Elves thought Dwarves were ugly and in the beginning, Elves did not even see Dwarves as sentient creatures and hunted them like animals.
The Eldar did not at first recognize these as Incarnates, for they seldom caught sight of them in clear light. They only became aware of their existence indeed when they attacked the Eldar by stealth at night, or if they caught them alone in wild places. The Eldar therefore thought that they were a kind of cunning two-legged animals living in caves, and they called them Levain tad-dail, or simply Tad-dail, and they hunted them. (Of the Quendi and the Eldar)
Even though they later figured out Dwarves were sentient, we have only have a few examples of friendship. It seems unlikely there was ever anything more, and if there was ever a Dwarven-Elven child, it certainly would have been notable and worthy of mention.
- There is one exception: Thingol the Elf and Melian the Maia. But we can suppose that Melian incarnated herself into the biological form of an Elf. :)
To the best of my knowledge we don't see it happen (or discussed) in any of the books, supplemental materials or reference guides so all we have is a bit of guesswork.
My gut instinct is no. Elves and human were created by Ilúvatar, the supreme being and creator deity. The dwarves were created by Aulë, the god of Metalwork and smithing, although later given independent spirits by Ilúvatar. As such, they're utterly different creatures, created at different times, albeit in much the same shape as the other races.
Although the films tend to present them as short, wide humans (in the same way that elves are thin effeminate humans), in the books they're definitely not similar to the other races.