These are pretty deep questions to which it's traditional to give sloppy verbal answers. Since it's so easy to find empty words about "wave/particle duality" in all sorts of sources, I'll try to write something a bit more careful and modern, at the risk of sounding technical.
Let' start with the simpler question. The Bohr model has almost nothing to do with modern physics. The Schroedinger picture of an atom captures a lot of the modern picture, especially when confined to the single-electron hydrogen atom. However, the full modern picture of what the electrons are doing in multi-electron atoms is given by quantum field theory, including quantum electrodynamics.
Now for the next question. The whole "wave-particle duality" has little to do with a modern picture of quantum mechanics. In the modern picture, everything (electrons, photons, etc.) is always represented by space-filling fields. Under some circumstances (called "measurements") large scale things (meters, brains,...) end up in different states depending on which way some little thing went. Our experience is always of a single state of large-scale things, for reasons I'll defer discussing for now. Therefore, instead of a whole spread-out wave (say of an electron approaching a CRT screen) your experience captures only one part, say the flash of light that would come from an electron hitting a small part of the screen. That's what motivated people at one time to say that the electron (or light, or whatever) sometimes behaved like a particle. However, there's no known way to consistently describe events like that except as the wave behaving as if it more more or less localized.
So is there any part of the current content of physics that's descended from the "particle" picture? Yes, these waves have a measurable physical variable which takes on possible values of 0,1,2,.... That's very much like counting classical particles, so we call that variable "particle number". However, frequently the waves do not have any single value of the "particle number" but are a superposition of different possible values. That's certainly true for an ordinary electromagnetic wave. That's not the sort of behavior you could imagine for things made up of little particle parts.
(published on 01/06/12)