You are talking about movie werewolves. There is no ancient legend or mythology that associates werewolves with the moon. Not even the most superstitious person before 1940 AD believed that humans turned to wolves when and only when the moon was full. There were superstitious people who believed that humans could turn to wolves and back. For some reason, astrologers never weighed in on the topic of werewolves. So the werewolves/moon connection was a false meme started in the movies.
The association between werewolves and the full moon was a meme that was first propagated by the first werewolf movie, 'Werewolf in London'. There was an radiation of lunar werewolves following another movie, 'The Wolf man'. Since then, there have been many variations on the idea of men who change to wolves at night during a full moon. No detailed analysis have ever been done on this fictional and nontraditional correlation between werewolves and the moon.
No entirely true. Firstly, the myth has roots in a lot of places, so there is no one ancient source to gather information from, but rather many writings from many cultures across Europe (Asian culture tends to have were-cats rather than wolves). While I agree that Hollywood movies have certainly have popularised and formulised what we now would consider the modern werewolf, the early 20th century movie writers drew on earlier traditions.
The earliest sources mostly seem to have no reasons for transformation, though it is sometimes emotions such as anger, or an ability or curse. Stripping naked and then donning the pelt, or a belt made from a wolf fur was also common. However, there was a belief from Russia that you could be transformed into a werewolf by sleeping naked outdoors under a full moon.
The most likely place that the current myth draws from, however, is the lore from Hungary and Balkans, the same place we get a lot of the modern (pre-Twilight) Vampire lore from. There it was believed that werewolves took form on nights of significant astronomical events, such as the winter solstice and also full moons. Thus to say that it was created by Hollywood is wrong, but since the texts from Hungary are 17-18th century, whereas werewolves, or at least shape-changers, go back hundreds of years earlier, then to say ancient texts don't include it would be correct.
The bad effect of silver on werewolves is also a movie meme. Evil spirits and monsters in ancient legend are often dispatched with iron. I don't mean just cutting or trauma. Creatures who are supernaturally evil just have to touch iron to be hurt badly. So a werewolf who runs when he sees iron is a mythologically correct monster. Running from silver is just a Universal Pictures fantasy.
Again, not entirely true. Yes, again Hollywood has popularised it, but it started to appear in German folklore in the 19th Century, so again, Hollywood writers drew on earlier sources. Iron certainly was the weapon of choice against the fey, but in the earliest sources of werewolves they tended to be totally vulnerable to standard weapons. One such tale tells of a woodcutter on his way home at night being attacked by a large wolf and striking off its front paw with his axe, only on his return home to discover his wife cradling her arm, a bloody stump where her hand should have been. Other stories have more unusual methods such as using weapons crafted from of dog or wolf bone, or finding the hidden skins and poisoning then with Belladonna (Nightshade) or Aconitum (Wolf's Bane). In one such tale the skins were rubbed with salt which then made the wearers vulnerable to weapons when they transformed.
Likewise werewolves could be killed by decapitation, drowning, asphyxia, or general massive trauma, such as dumping a boulder on them.
Using cold iron or iron forged weapons isn't one I have seen a lot of in the tales I have read, though I suspect you are generalising from other supernatural types to include werewolves, not going by the tales told.
