First of all, concerning the question
Why the atom is not physically in the full-Hamiltonian eigenstates which we can't calculate exactly,
akhmeteli correctly writes that this question and all analogous questions and proposals saying "a physical system must always be in some preferred basis vectors" is a fundamental misunderstanding of quantum mechanics. A basic universal postulate of quantum mechanics is the superposition principle: whenever a system is allowed to be in certain states, the same system is always allowed to be in an arbitrary superposition (complex linear combination) of these states! So it's manifestly wrong to say that an atom is "always in an energy eigenstate" or "always in one of the basis vectors of another basis you randomly picked".
It may be hard to prepare a large enough physical system, like Schrödinger's cat, in a general superposition and it is even harder for many people to imagine that it's possible at all. However, in principle, it's always possible; it's what quantum mechanics guarantees. It just doesn't allow atoms or other physical systems to be "constrained" to "physically be" in a previously chosen basis.
Now, concerning the emission of radiation, first we must ask: What does it mean that there was an emission? It means that the energy carried by the radiation is different (higher) at the end than what it was at the beginning. It's possible to reformulate this proposition (which may be right or wrong after a particular time interval and in a particular physical system) in terms of photons: the emission of radiation occurs when the number of photons increases. These equivalences are completely general; they encode what the very phrase "emission of radiation" means.
We should ask: Why does a quantum mechanical theory allow the number of photons to be increased? It's because the Hamiltonian has the form
$$ H = \sum_{i} \hbar\omega_i \cdot N_i + \sum_i \hbar a_i \cdot d_i + \sum_i a^\dagger_i\cdot d_i + \text{non-photon-related} $$
The first term which is a sum over one-photon states just counts the total energy stored in the photons: each photon contributes $\hbar\omega$ to the total energy. This first term preserves the number of photons: an $X$-photon state is evolving to an $X$-photon state. Note that $N_i=a^\dagger_i a_i$ where $a_i,a^\dagger_i$ are properly normalized annihilation and creation operators.
The two following terms containing the letter $d$ change the number of the photons and they're needed for the emission and absorption processes because, as argued previously, emission or absorption is, by definition, the same thing as the change in the number of photons. So these terms contain either the annihilation operator or the creation operator without its Hermitian conjugate partner. I chose the letter $d_i$ for the dependence of these terms on the quantum number of the atoms – this $d_i$ acts on the atoms' degrees of freedom etc. and in some approximations, the most important piece is expressed by the atoms' electric dipoles etc.
It's important to emphasize once again that I am not saying that the number of photons $N_i$ must always be sharply defined in any state allowed for the physical system. It's not true at all; all superpositions are possible. We're just trying to calculate a particular meaningful answer – the probability of emission – and we find out that the probability of emission depends on the initial state, especially on the number of photons in various states. So we're considering these initial states with a certain number of photons because they're a part of the formulation of the question. One may also calculate the probability of emission for a general initial state that is a superposition of states of different values of $N$ but in that case, one must be a bit more careful in defining what the term "emission" really means.
The emission and absorption processes change one particular number of photons $N_i$ for one value of the index $i$. The Hamiltonian above shows that even if $N_i=0$ in the initial state, there is a nonzero probability that we will get a $N_i=1$ final state: so the number of photons may spontaneously jump even if it is zero to start with.
Albert Einstein was the first one who figured out the most general argument why it's so: the absorption has to be "stimulated" because one must absorb photons that actually exist to start with. No photons means nothing to absorb. But the probability of absorption is a function $f(N)$ of the number of photons in the initial state $N$ which therefore gets changed to $N-1$: one photon is subtracted.
Now, the microscopic probability (classically) or probability amplitude (quantum mechanically) must be the same for the time-reversed (more precisely: CPT-conjugated) process. The time-reversed process to absorption is emission. But the initial state gets mapped to the final state and vice versa.
Because the probability of absorption goes like $C\cdot N$ where $N$ is the number of photons in the initial state and because the number of photons in the final state is $N-1$ and because this $N-1$ becomes the number of photons in the initial state of the time-reversed process, we see that for $N-1$ photons in the initial state, the probability of emission is proportional to $N$. In other words, for $n=N-1$ photons in the initial state, the probability of emission goes like $n+1=N$. In the form $n+1$, the term $n$ corresponds to the stimulated emission and the term $1$ corresponds to the spontaneous emission.
