Finrod, the eldest son of Finarfin, brother of Galadriel, and founder of Nargothrond, earned this name from the Dwarves. When he discovered the caves of Nuluk-khizdîn, he decided to create a kingdom similar to that of Menegroth. Nuluk-khizdîn was first inhabited by the Petty-Dwarves, who had expanded and widened the natural caves for their own use. Finrod did the same thing, but to a much greater extent, and for this process the Dwarves named him “Felak-gundu”, the “Hewer of Caves”. Finrod used the name thereafter, and modified it slightly to suit the speech of Elves it became “Felagund”. This is one of the very few examples of Khuzdul to have been borrowed and used in the Elven languages.
Felak is "hewer", and Tolkien describes it as "a tool like a broad-bladed chisel, or a small axe-head without half, for cutting stone." It sounds like a smaller tool held with one hand, perhaps driven into stone using a hammer on the butt-end. Felak is derived from the verb felek "to hew rock". Tools or agentive nouns then may, at least in some cases like here, be derived from verbs just through a change in the vowel pattern. For comparison Arabic has katab "to write" vs. kâtib "writer". Felak can also be used as a verb meaning "to use a hewer".
Because felak is the first word of the compound, it should be in the composition form. There is a chance that the indefinite may be something other than felak, like maybe fêlak or felâk, but because Tolkien gives this form with the translation "hewer" and no further comment, I think it's best to be conservative and assume this as the indefinite form as well. It is probably also singular and nominative.
Felak then is "hewer" and is singular, nominative, composition.
Gundu is translated as "cave" or "underground hall". Like felak, it is derived from a verb, which is gunud "to delve underground, excavate, tunnel". This implies, along with the manner of how Nargothrond was formed by manually enlarging whatever caverns were already there, that these words are not referring to natural caves. Instead, they are at the core of how Dwarves seem to fashion most of their homes, which is to tunnel underground. The same element is apparently seen in Gundabad, which would be the underground hall from whence Durin I awoke. There it appears without the -u ending. Looking for an explanation, we find one readily available in Adunaic, which is the objective genitive. This is two nouns placed in a compound word, where one of them is derived from a verbal root and thus has an associated action (such as "hewing"), and the other is in the "objective" case to indicate that it is the object of that action (here, the "cave" is being hewn). The objective case in Adunaic is also formed by a -u suffix (sometimes infixed), just like we see here and also in Uzbad Khazad-dûmu and mênu (which is in the accusative case). Thus, Felak-gundu is probably "hewer of caves", where the cave is being hewn, compared to Felak-gund: "a hewer at or in a cave", “a hewer that comes from a cave”, or “a hewer that belongs to a cave”. Another interpretation for the standard construct would be "X made of Y", but saying “a hewer made of caves”, in this case, obviously makes no sense. For another pair of words, it could.
Based on this, I conclude that gundu is "underground, excavated hall" and is singular, accusative, indefinite.
I see Felak-gundu, as a whole, to be an objective construct compound word consisting of noun-noun word order.
Additionally, Tolkien provides the form Felag-gundu. This shows that unvoiced stops, when place next to their voiced counterparts, probably become voiced as well. This may or may not be indicated in the orthography. Assuming dûm and tumun are indeed from the same root, then we would have Khazad-tûm assimilating to Khazad-dûm. This shows that the voicing assimilation occurs both forwards and backwards.
The Silmarillion, index, pg 330
The Peoples of Middle-earth, pg 352