As a matter of fact, I found the source, although curiously it does not say exactly what people quoting it seems to imply:
Some slightly better informed claims directed to article 166 of Turkish penal code of 1911. As my knowledge of pre-Ataturk legal Turkish jargon is a bit lacking1 and all the sites claiming against gun control failed to provide the actual text, I had to rely in a horribly formatted translation of it from 1913 which I will try to make sense from2:
Whoever, without obtaining permission from the department concerned, manufactures within the Ottoman territories gunpowder or other explosive substances 1 or prohibited weapons 2 or cartridges for them or imports into the Ottoman territories from foreign territories gunpowder or other explosive substances 1 or prohibited weapons 2 or cartridges for them, or becomes a medium for this sort of smuggling, or transports 3 or imports 4 from one place to another place within the Ottoman territories smuggled gunpowder or other
explosive substances 1 or prohibited weapons 2 or cartridges for them is, in addition to the confiscation of such, put in prison for from two months to two years, and a fine of from five Mejidieh gold pieces 5 to fifty Mejidieh gold pieces is taken. Those who, without permission, carry or sell such
prohibited cartridges, weapons, 2 gunpowder or explosive
substances 1 are also punished with imprisonment for from
one month to six months and by taking a fine of from one
Mejidieh gold piece to ten Mejidieh gold pieces.
What is interesting is what seems to be the footnote by the translator:
For the purposes of 6 the Penal Code " prohibited weapons " mean
generally State or military weapons 2 and revolvers of which
the barrels are more than fifteen centimetres.
So, what is clear:
It only affects a limited range of weapons (military of revolvers with barrels longer than 15 cms).
It just defines that a permission is needed but does not explain the requirements for obtaining one. Was it easy or difficult? Was it given or denied on a basis of race? There is not enough information about that.
It is worth noting that the 1864 version does not talk about permits but about regulation for storing powder and the damages that it could cause in the case of fire (i.e., more of a hazard regulation than gun control). The 1903 version again asks following regulations for production and totally forbids (no possibility of government permit) its importation.
As a side note, the 1910 version was already pretty similar to the final version (so it would make more sense to talk about the 1910 law).
There are reports that, prior to the beginning of the Armenian genocide, Turkish officials demanded Armenians villages to surrender their weapons. So, either they were legally or illegally held weapons, it is clear that the law did not prevent the Armenians having weapons.
In a more personal opinion, all of the above makes me think that the article is directed more to the fabrication and distribution of weapons than to their ownership (again, this is just a personal opinion, feel free to think otherwise -or better yet, provide an example of the article being used to punish someone for carrying a weapon-).
As for the questions:
I would say "No" to the claim that the intent of the law was to perpetrate the genocide.
My main point is the one already explained in comments; the Armenian genocide has been widely studied and I cannot find any claim about this law being linked to it. It would be a quite strange omission if there existed any evidence to show such a level of premeditation and the people studying the Genocide failed to highlight it.
Note that in 1908 and 1909, Turkish military reforms aimed at including non-Muslim individuals to serve in the Turkish army... If they were already planning to massacre them later the last thing they would want to do was to give them military training3.
Also, as stated above, if it was meant to disarm the Armenians then it would was a notable failure, or else it would have made no sense forcing them to disarm in 1915.
The second part of the question cannot be truly answered. To answer that, one would have to be able to "see" an alternate universe in which there was not such law (we have an "alternate history theory" reason to close that kind of questions). Just a few thoughts (even if it slightly off-topic):
Almost every country in the world has gun control laws, to some extent (yes, including the USA). There is nothing easier to wait for something bad to happen in any country and claim "because of gun control laws", there is no way to prove or disprove that (no "alternate universe" available). You might as well say that the Armenian Genocide (or others) would not have happened if we were living in Mars, or that gun control laws are the cause of cancer.
In many cases of genocide or oppression, the main advantage of the oppressor is not in brute force but in organization. They subdue people in small numbers because they can concentrate their forces. This was how politics worked in the old ages (Roman Empires, Germanic and Muslim invaders, Spanish Empire in America; none of them had a technological advantage enough to explain how they dominated subjects that greatly outnumbered them).
Note that in 1915 Turkey was fighting WWI and many of the Armenian men capable of fighting were already conscripted in the Turkish army(in unarmed labour battalions), making the difference of having available weapons even less significative.
1Please forgive me.
2Note that in the text referred there are several version of the same article as it was successively amended, the one above is the last one.
3Immediately prior to the genocide, Armenians serving in the Turkish army were moved to unarmed labour battalions, but that was way later than the law had been approved.