Are integers such as n=1,2,3 etc in Bohr's atomic theory, exactly whole numbers or just very close to being whole numbers?
They are intended to be exact integers. Please note, however, that Bohr's model with electrons moving in classical circular orbits, and Sommerfeld's version with elliptical orbits, are incorrect and obsolete. They were supplanted by the quantum-mechanical model using the [itex]\Psi[/itex] function, developed by Schrödinger and others in the 1920s. You should consider the Bohr-Sommerfeld model as being of historical interest only. If you insist on burying yourself in the details of the Bohr-Sommerfeld model, you'd better be prepared to set them all aside when you start to study the Schrödinger model.
Bohr's model uses the idea of standing waves, with the wavelength determined by debroglie's relation. the wavelength must be an integer multiple of pi*r for there to be standing waves. n*pi*r = wavelength = h/p is the main idea here. You can solve for velocity: v = h/(n*pi*r*m), then substitute this velocity into the force equilibruim equation: m*v^2/r = q^2/(4*pi*epsilon*r^2), and solve for r. You can see that you'll get r in terms of some value times a function of an integer. That is the gist, but I might have messed something up...
Bohr's original model (1915 or thereabouts) did not use waves for the electron. He simply postulated that the electron moves in a circular orbit whose orbital angular momentum mvr can have the values [itex]nh / 2 \pi[/itex], with n = 1, 2, 3, ... About ten years later, de Broglie came along with his idea of using a wave to represent an electron, as you describe, to explain Bohr's quantization condition. But he still thought in terms of circular or elliptical paths, whereas Schrödinger's wave function spreads out in three dimensions, and are more like standing waves of sound in a spherical cavity than waves going around a ring.
Yes, they are intended to be exact. They are derived from the Balmer equation, and subsequently Bohr reproduced Balmer's equation using an electrostatic theory. DeBroglie later suggested that these quantum intergers represented the half spin of an electron in its orbit. Here, you have an interesting situation. Bohr's theory, while generally regarded as flawed, is still contained within most pertinent texts. The reason is that it provides the student with a highly visual representation of how electrons make transitions between quantum states. The problem with the theory is that it works, it does explain empirical data almost exactly. Unfortunately, Bohr was never able to describe how an electron exists between quantum states. Specifically, the transition was suppose to occur instantanenously, but nothing was suppose to move faster than the speed of light. Further, Bohr was never able to extend his theory to more complex atoms that were not Hydrogen-like, i.e., it only worked for Hydrogen and other atoms stripped of all electrons except one. Yes, quantum physics used some of the underlying ideas of Bohr's model. And yes Bohr won a nobel prize for his work. It is also true that he accepted that his theory was wrong and helped to further quantum physics. Then again, there could be a global conspiracy to hide how the atoms really work- shrouding the science of atoms in an almost impossible probability theory- ha ha. You should also note that quantum physics has only one viable solution, just as Bohr- only for Hydrogen. I believe that it adapts this equation to approximate the transition of electrons in other atomic settings. Also the value n is still used in quantum physics, in modelling the electronic configurations of the atoms.
Ironically Bohr with 25 years were awarded NObel prize in Physics by this 'model', perhaps there were some 'bribe' with the jury...for my this and photoelectric effect are the most 'unworthy' (they didn't deserve winning any nobel prize) of all history, if physics were today as easy at it was in 1900 (and until 1930) anyone of us could be 'Einsteins' don't you think??