\$N_A\$ and \$N_D\$ depend what dopants you apply to the sample. They don't have to obey any particular laws.
The n and p concentrations that result from the dopants depend on some physical laws. For non-degenerate doping we usually use (and you can solve your problem with) the law of mass action:
$$n_0p_0=n_i^2$$
Once you've worked out the \$p_0\$ boxes in your table, you then can figure out the unspecified dopant concentrations by knowing that the dominant dopant is providing all of the majority carriers, plus it's providing or capturing enough electrons to ionize the other dopant (because of the assumption of all dopants being 100% ionized).
The electron states in the band gap are localised, whereas the states which contribute to the bands are not.
The electron states in a solid are not simple, there's a lot of non-trivial quantum mechanics going on. The electron states in a free atom are localised around the atom - the electrons in those states can't leave the atom without a lot of energy, so they can't conduct anything. When you pack lots of atoms together, the surrounding electron states overlap and mix. Which states mix with each other is dictated by the energy: similar energies means more mixing. You end up with a new set of states which extend over the whole block of material. If the material has a periodic lattice, these electron states group together into bands.
Every state in a band has some velocity (called a Fermi velocity) associated with it, and an electron in that state can be thought of as moving through the material with that velocity. The Fermi velocity of electrons in the conduction band is very large, but because the electrons are all going in different directions, there is no net current. An applied electric field moves some electrons from states which were going in one direction, to states going in the other. In a metal, one of the bands is part full, so there are plenty of nearby states to move electrons into. In a semiconductor, there is a gap between a full band and an empty one so it's much harder to push electrons into the higher band.
When dopants are added, they don't form a nice periodic lattice and they are much more spread out than the silicon atoms that host them. This means that the electron states around the dopant can't mix with states from other dopants to form a band. Since the energy levels of the dopant states are different from the silicon states, they don't mix (much) with them either. Instead, the electron states are localised around the dopant, much like the states around the free atom. An electron in that state can't conduct in the way one in a band can. It either has to jump up into the band, or jump to another nearby dopant. The former happens in semiconductors, the later is known as incoherent transport, and appears in some other materials.
I'm not sure how well I've explained this, but if you don't get a clear answer here, you could try the physics stack exchange. This definitely feels more like condensed matter physics than electrical engineering!
Best Answer
Non-organic semiconductors are used in crystalline form. The definition of a crystal is not that it looks like a gem, though that is a common result, it's that the constituent atoms are arranged in a regular pattern known as a lattice. A crystal of silicon forms a diamond-cubic lattice structure like this:
If you look carefully (and understand that this is just one cell; it replicates on all sides) you'll notice that each atom has 4 bonds. Each bond involves one electron from each atom. In normal silicon, some electrons do dissociate from their parent atoms and leave holes behind (this is \$n_i\$, the intrisic carrier concentration). However, the normal pattern is for a bond to more-or-less permanently link an electron and a hole.
The reason that trivalent and pentavalent atoms are acceptors and donors is that this bond structure is maintained when impurities replace atoms in this lattice. If you add a trivalent atom to this structure, you still require four bonds but you only have three valence electrons to work with. This is an acceptor. Each trivalent acceptor atom (in a unit volume) increases \$N_A\$ by one. Silicon cannot be a donor because it only has four valence electrons; if it donates an electron to fill this bond then that leaves a hole on the silicon atom. If you add a pentavalent atom to the structure, four of its electrons will form bonds but the fifth cannot form a bond. It becomes a donor, and each pentavalent donor atom increases \$N_D\$ by one.
If you added an impurity with two or six valence electrons, the same thing would happen, but two electrons or two holes would be added per atom. However, this is more likely to cause a break in the crystal structure and isn't really very common in industry as far as I know.