The two questions you are asking are quite dissimilar.
Well, very clearly the assumption seems to be that uncus has a silent /k/ and that Latin /k/ reflects PIE *k in this case, and that g reflects a voiced consonant as well as most of the other cognates that mean "nail". The Laryngeals h2 and h3 are informed by other languages, too, Greek o for nail implying h3. Overall, unguis seems quite certain, according to the many references given, though there are dialectal Greek forms (see Beekes), as well as Hittite showing k (No g is known in Hittite so that shouldn't be surprising). Given the sound laws, that should be mostly a matter of arithmetic.
For uncus I am not sure. It is derived via an intermediate full grad form, from *h₂enk-
from From Proto-Indo-European *h₂ónkos (“hook”). Cognates include Ancient Greek ὄγκος (ónkos) and Sanskrit अङ्क (aṅká). [via uncus].
The semantics of all three agree, and then sound laws are sought to match. anka indicates *h2 (thus dubbed the a-coloring laryngeal, though I'm not sure what it yields in the environment of o, apparently a u). Greek ongkos on the other hand indicates *h3 as far as I can see, and even contains a g. (at that point it's notable that ny and gamma can look perceptively similar if the lower stroke of small gamma is not pronounced). We further find at *h₂enk-:
"*h₂enk-os or *h₂n̥k-os", Celtic *ankos "bent", deriving Celtic *ankotos "hook, paw";
Germanic *angulaz, I think you can see where that is going;
Old Armenian angł ~ ankł ... "From Proto-Indo-European *h₂enk-u-l-, from *h₂enk- (“to bend”), referring to the bird's curved beak.", that means "vulture" or "handle of a pot or basket"; compare that with German Henkel, "handle of a mug, can, etc." supposedly from henken, akin to to hang, that has its own PIE root, looking similar to Henker, "hang-man, executor", cp. "jemanden an den Haken kriegen" (to catch somebody, to hook someone)?
etc. etc.
While the reconstructions are imperfect, it's reasonable to assume that the language was diversified.
There are copious reconstruction that may be grossly glossed curved, bent, cp. Ecke, Angst, Enge, Knick, anchor, horn, thorn, corner, acute, nose, peak indeed bent, bow themselves ... if that's your thing I'm sure there are more the further you want to loosen the semantics, In the end one has to concede that PIE is meant to reconstruct a full blown language. Connecting the roots below that level is a different task and for the most part very much uncertain. So it's possible that you have stumbled on an coincidence.
As the saying goes: If all you have is a hammer, everything looks like a nail. Be careful not to hit your finger-nails.
So there's one more observation to make. Ger Unke (a kind of frog?) leads to *h₂éngʷʰis (“snake”). Whereas snake leads to *sneg- "to crawl; a creeping thing" which has been compared to *(s)neh1- "to sew, ..?" (whence needle), see e.g. [Kloekhorst 2014:4](*sneg- (http://www.kloekhorst.nl/Publications.html). I'd add that snake teeth make good needles; And that needles may be hook shaped. Or a finger nail?
One more: The root *h2ek- "sharp" is so general, that it might as well connect both your words. Given ex, *eghs-~*eks- (and other alternatives), plus idioms like "to peak out" a whole paradigm seems to crop up. Words meaning to cut, break, divide, share are rather numerous, too, for example.
Tangentially: I see no satisfying reason why wood work nails should be akin to finger nails, other than being sharp.
I'm afraid, in the end this the nature of this forum seeking definite answers is not well suited to the question at hand. So, I could go on and on, but without providing a definite answer according to the standard, so let's cut it short.
