Why are they comparing state value function to action value function?
It is because $v_{\pi}(s)$ and $q_{\pi}(s,a)$ measure the same quantity at different stages of the trajectory. By comparing the values at the same $s$ and modifying how $a$ is selected, the proof makes assertions about how that choice impacts the value.
It is important to recall that $v_{\pi}(s)$ measures the expected future reward when starting in state $s$ and following policy $\pi$, and that $q_{\pi}(s,a)$ measures the expected future reward when starting in state $s$ and taking action $a$, thereafer following policy $\pi$. When $a$ is chosen using a deterministic $a = \pi(s)$ then $v_{\pi}(s) = q_{\pi}(s,\pi(s))$
Isn't it obvious the above equation might hold true
The inequality (it is not an equation) does strictly hold true, because the equation $v_{\pi}(s) = q_{\pi}(s,\pi(s))$ is true. However, that is not terribly useful, it doesn't prove anything new.
What is interesting is if you change the decision for $a$
(provided we select the best action among the possible actions)
You cannot do that. The policy decides the actions, by definition. Selecting a better action according to the action value function, and showing what that does is precisely what the proof is showing.
since then the equation will change to $q_\pi(s, \pi(s)) \geq v_\pi(s)$
$q_\pi(s, \pi(s)) = v_\pi(s)$ - yes your inequality holds, but is equivalent to not changing anything.
and we know $v_\pi(s) = \pi(a|s)q_\pi(s, \pi(s))$?
Here you have switched to using a non-deterministic policy and a deterministic policy in the same statement (you also have not defined $a$ properly), making your equation badly formed. The correct form would be:
$$v_\pi(s) = \sum_{a \in \mathcal{A}(s)} \pi(a|s)q_\pi(s, a)$$
This is not a relevant form for the initial policy improvement theorem. However it does become relevant when the theory is adapted to show improving $\epsilon$-greedy policies later.
NOTE on Convention: As per the convention of the book goes, I think they are using rewards for state to action to state transition sequence rather than state to action transition.
That is not a convention in the book. In the book, the reward distribution is generally given as joint distribution with state transition in $p(s', r | s, a)$ - this is a very generic approach that models any discrete MDP, regardless of how the rewards are described as associated with any of $s, a, s'$ or a random factor.
Possibly the convention you are referring to is labelling the immediate reward as $R_{t+1}$, so that a small section of trajectory starting from state $S_t$ might be $S_t, A_t, R_{t+1}, S_{t+1}$ . . . some other texts will use $R_t$ here. However, that is not relevant to the policy improvement theorem, it would hold the same under either convention, just with slight relabelling of the reward indices.
