I think the question title is clear. What happens when other conducting metals are also around the rod? How could you explain intuitively?
In general, the charge induced on the conductor will not have the same magnitude as the charge on the rod.The reason is that not all field lines from the rod will end on charges on the conductor but go to infinity or to other nearby charges. The closer the rod is placed near the conductor the higher the induced charge on the conductor will be.
Assuming everything is within finite distances, yes the charge will always be induced. Take an arbitrary shape and give it an arbitrary distribution. If we step back, 'far enough,' the field will kind of look like a charged particle of the total charge, $q$, of the mystery object. Now let's bring in a conductor of your (finite) choosing from far away. What happens to it? Well, it sees a charged particle, so it induces that charge. Bringing it closer and closer doesn't change the fact that there's an overall charge $q$ right there. The charge distributions are whole other story! The overall charge, it's just this.