Ok, it looks like I missed something... How do we know that final -v becomes -w in Sindarin? 

And what's _# supposed to mean?

Unless I'm missing something, one finds -w only after a long ā, as in raw 'wing' < rāma (PE17:63), so what really happens is that /v/ gets lost in coda after /u/, *rauv > raw. After a short vowel /v/ is preserved, see alaf 'elm' (PE17:153).

The notation _# means 'in final position', _C would be before the consonant C. It's a formalism I don't really understand - it doesn't save much writing effort and makes it less readable to others...

Sorry, I probably should have written it out in English. Also, I made a small mistake. It should be v>w /V_#

In Sindarin, /v/ became a /w/ when it is found between a Vowel and the end of a word. (The underscore stands for the letter I was talking about, and the pound sign stands for the end of a word. The V is shorthand for "any Vowel".)

Evidence: (I'll use Sindarin/Noldorin VS Quenya because that's easy)
Q. lómë - S. dû (-w deleted in presence of /u/)
Q. lúmë - S. lû
Q. tuima - N. tuiw
Q. hráva - S. rhaw (see also: rhavan)
Q. caima - N. caew
Q. laive - N. glaew
Q. haime - N. haew
Q. himya- - N. hîw
Q. yáve - N. iaw
Q. quáme - N. paw
Q. ráva - N. raw
Q. roime - N. ruiw

Tolkien, even in the Noldorin phrase, was considering this idea, but applied it inconsistently - or so it seems. I'm not sure. I do know that it's described in PE17:133 for final m.

It seems then that final /v/ became /w/ after the vowel /i/ and disappeared after the vowel /u/ but remained after other vowels. Thus Noldorin lhâf should probably be Sindarized as lâf. 

Turned into /w/ after high vowels... interesting idea! Not just high vowels though, perhaps only high long vowels/diphthongs? This change would have to come before AI and OI became AE and OE, but after /o/>/au/ and /e:/>/i:/.
What about Fladrif? Why does this /v/ stay around?
Could the name have been formed before this change, leaving the /v/ behind a short /i/?
Or perhaps the missing element is Stress, not the number of morae the syllable takes. That may actually be simpler. But what about Orome-Araw?

This stream of conscious linguistic babble brought to you by 8 AM and a looming interview. I hope it's not too unintelligible.

The length proviso covers all the words in your list (including hîw, which derives from ¤khīmā) and Araw. As for Fladrif  there seems to be nothing about its etymology on record, so for all we know its i  might always have been short.

But if it was always short, wouldn't it have become an E?

Maybe it was long, became part of the name, lost length, then the v>w change happened - meaning that the conditions for the change no longer existed.

Short I became E only when subject to a-affection. Cf Q rimba : S rem 'numerous'; Q rimbe : S rim 'host, great number'. 

Really? I hadn't heard this... We need to build a redo of the Etymologies, tracing each word through its changes  and make a massive collection of all the phonetic changes... all attempts at it that I've seen are incomplete. Tolkien's work is too much to analyze for one person alone.

For vowel mutations, see the article by Bertrand Bellet:
http://www.tolkiendil.com/langues/english/i-lam_arth/vowel_affection_sindarin_noldorin

But back to the topic: Already in the Etymologies we find N. nîf < nībe (NIB-) and N. rhîf corresponding to Q. ríma (RĪ-). In later Sindarin, there is also îf 'cliff' < īb- (PE17:92) (rhîw is an uncertain case, I find to primitive form given), so it would appear that N. hîw < khīmā is the oddball here.

After diphthongs we have N. hnuif < *sneumi (SNEW-) with preserved /v/ in contrast to rhui(w) < *roime (ROY1-), tuiw < tuimā (TUY-) and others.
For later Sindarin both patterns are attested as well, see S. oew < okma (PE17:170) and S. lhoew, Q. hloima (PE17:185), but S. raef, Q. raima (VT42:12).

From an external perspective it seems that this is one of those sound shifts Tolkien didn't have a fixed, immutable conception of. From an internal perspective one could explain the variations by dialect loans.

In Araw, we have /auv/ > /aw/ both according to WJ:400, as well as PE17:153 - this development does seem to be a fixed conception, so it's unproblematic.

But it's a good thing that you brought this up, since I've realized that N. rhîf was regularised to S. *rîw in our wordlists which seems completely unjustified.