The nice clean energy level diagram, with the valence band below, the conduction band above, and the well defined band-gap in the middle, is something of an idealization. It's approximately what you'd get if you had a perfect lattice, and only the exact dopant atoms in a nicely even distribution.
Any number of things are going to fuzz up those clean lines. Mostly impurities will be the culprit. The intended dopant atoms are impurities in the lattice that happen to create energy levels near enough to the conduction band that thermal variations are enough to allow electrons to escape from them into the conduction band (N-type) or settle into them and leave a 'hole' to act as a charge carrier (P-type). However, it's just not possible to prevent other, unwanted impurities from getting in, and these can put energy levels anywhere in the gap. Thermal variations aren't enough to let electrons out of (or into) these kinds of levels, hence the name 'trap'.
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
Thermal voltage (\$\sim 25\,meV\$) is the average kinetic energy of particles in gas. It's not the total energy of each and every electron.
The actual energy distribution among the electrons is described by Fermi energy and temperature via Fermi Dirac distribution. At zero temperature, all electrons have energy less than Fermi energy. At non-zero temperature, some electrons have energy greater than Fermi energy.
So at any any temperature, the probability that an energy level is occupied is given by Fermi Dirac distribution, \$f(E)\$. The density of available states in an energy interval \$dE\$ at any energy is \$D(E)dE\$. Where \$D(E)\$ is the electron density of states function. Then the total electron concentration can be calculated by,
$$n=\int\limits_{E_C}^{\infty}f(E)D(E)dE$$
\$E_C\$ is the bottom of conduction band and \$D(E)\propto\sqrt{E}\$. Few approximations are used to do this integration. After integration, $$n=\sqrt{N_CN_V}\exp\left(-\frac{E_g}{2k_BT}\right)$$
where \$N_C\$ and \$N_V\$ are material and temperature dependent parameters called as the effective density of states and \$E_g\$ is the bandgap of the material.
For silicon at room temperature, the bandgap is \$E_g=1.1\,eV\$ and the effective density of states values are in the order of \$10^{19}\,cm^{-3}\$. This will result in an intrinsic carrier concentration \$n_i\sim 10^{10}\$. The exact values can be obtained from books.