How is the spontaneous decay rate of an atomic level calculated? I am trying to understand the transition rate $\Gamma$ between two atomic state levels when there are multiple decay channels involved. Let's say I am considering the spontaneous decay from an upper $P_{3/2}$ level to only one possible lower $S_{1/2}$ level. The upper level has $4$ sublevels corresponding to each of $m_J' \in \{-3/2, -1/2, 1/2, 3/2\}$. Similarly, the lower state has $2$ sublevels corresponding to $m_J \in \{-1/2, 1/2\}$.
Let's assume I am only interested in Electric Dipole (E1) transitions. When I say I am calculating the decay rate $\Gamma$ of the upper level to the lower level, what am I doing? I am confused between two different ideas:
a) I am calculating the rates for each transition $|{P_{3/2}, m_J'}\rangle \to {S_{1/2}, m_J}\rangle$ allowed by E1 rules and averaging them to get $\Gamma$.
b) I am calculating the rates for each transition $|{P_{3/2}, m_J'}\rangle \to |{S_{1/2}, m_J}\rangle$ allowed by E1 rules and summing them to get $\Gamma$.
I believe the correct idea should be a), since that makes intuitive sense, but I have seen idea b) in textbooks and other references, which confuses me.
 A: For each upper sub-level, you sum the transition probabilities into all available lower sub-levels (taking the selection rules into account), then you average these sums over all upper sub-levels.
As an analogy, imagine a particle in a box. If you have two holes in the box, the probability of the particle escaping is twice as high as with just one hole (hence you sum the lower sub-levels). And if for some reason the particle can be in different states that affect the probability to go through a hole, you have to average over these options (because it can be only in one state at any one time but you don't know which). It would be different obviously if you could prepare the atom so that you have only one upper sub-level occupied. In that case you just calculate the transition probability for that sub-level. Generally speaking, the upper sub-levels could be occupied with a different probability, so you would have to do a weighted average in this case.
A: I will try to give an answer explaining the solution from some basic principles of sponteneous decay.
For E1 transitions, the central quantity appearing in the decay rate between two states is the dipole moment. More precisely, the expectation value between the two states of the dipole moment operator:
$$\Gamma_{e \rightarrow g} \propto |d_{e \rightarrow g}|^2 = |\langle e | \hat{d}|g\rangle|^2 \,,$$
where $|g\rangle$ ($|e\rangle$ is the ground (excited) state of the transition. For an electron in an atom, we have $\hat{d} = -e\hat{r}$. The decay rate is then calculated via Fermi's Golden rule or Wigner-Weisskopf theory.
When we have multiple states and transitions between them, there are two questions we can ask, as observed in the question. I am going to reformulate them here, to show that each of them has a physical meaning (note that these do not correspond to a) and b) in the question, but are two different scenarios).

*

*At what rate does a given excited state decay? For this question, we want to know how fast that excited state is depopulated by spontaneous emission. Since it can go via all the channels and end up in each lower state, we have to sum over all decay channels, such that
$$\Gamma_{e, \mathrm{tot}} = \sum_g \Gamma_{e \rightarrow g} \,.$$
For more complicated situations, there may also be multiple decay channels between the same states (such as higher-order multipole or non-radiative decay), such that summing over the transitions instead of the ground states may be appropriate.

*However, we could also be interested in at what rate does the excited state decay to a given lower lying state? Then we are interested in $\Gamma_{e \rightarrow g}$ on its own.

To answer the specific question in the OP...

When I say I am calculating the decay rate Γ of the upper level to the lower level, what am I doing?

With "decay rate", we usually associate how quickly a given state decays. This situation corresponds to 1. Note that you do not average over the different channels, you sum them (option (b) in the question). The reason simply being that when you sit in the upper state, the electromagnetic environment couples you to all the (E1 allowed) lower lying states, so you can spontaneously decay into each of them.
Note that for degenerate or dynamical systems, the excited state does not necessarily have to be an eigenstate of angular moment etc., but can also be a superposition of eigenstates.
For more details, take a look at the derivation of Wigner-Weisskopf theory or Master equations in quantum optics. They will give you an intuition of what the decay rate actually means and how it acts dynamically to change the state population.
Also note that for angular momentum eigenstates, there are often relations between the different transition dipole moments via the Wigner-Eckart theorem. So often the sum over the decay channels can be expressed in terms of one of the dipole moments and a bunch of Clebsch-Gordon coefficients. Note that one has to be careful to correctly apply the Wigner-Weisskopf formula accounting for this.
