In some willows, there is a clear distinction between floriferous buds (out of which the catkins grow) and vegetative buds. I have observed this phenomenon myself on numerous occasions (in fact, I can ID several species and hybrids on floriferous buds alone). However, I never took it further and there are things I still don't understand. What about those vegetative buds. What grows out of them: twigs with leaves on them out of some and single leaves out of others? And would there be any noticeable difference between these two types of vegetative buds (if they both existed)?
To answer the question in you title, the leaves appear both on new and old growth. In addition to vegetative and floral buds, there are also dormant buds that activate themselves on old growth after leaves or branches are cut. You can read here a list of buds.
For the following questions in the text I think the answer is different for every species.