As in the previous answer, the equality $\alpha + p_6 = 1$ can be rewritten as
$$\sum_{n=3}^\infty g(n)p_n=1,$$
where
$$g(n):=(n-6)^2+1(n=6)\ge1(n\in\{5,6,7\})+4\times1(n\notin\{5,6,7\}).$$
Therefore and because $p_n>0$ for all $n\ge3$, we have
$$\sum_{n=3}^\infty g(n)p_n>\sum_{n=3}^\infty p_n=1,$$
so that the equality $\alpha + p_6 = 1$ is always false.
It is also clear now that, for the equality $\alpha + p_6 = 1$ to hold approximately, we need the probabilities $p_n$ for $n\notin\{5,6,7\}$ to be negligible compared with $p_n$ for $n\in\{5,6,7\}$. Since
$$p_n=e^{h(n)}/Z$$
with $h(n):=-c_1n-c_2/n$ and the mean $\sum_{n\ge3}np_n$ is $6$, we see that (for the equality $\alpha + p_6 = 1$ to hold approximately), the $p_n$'s hence should peak near $6$, that is, $h$ should peak near $6$. Since $h'(6)=-c_1+c_2/6^2$ and $h''(6)=-2c_2/6^3$, we see that $c_2\approx 36 c_1$ and $c_2$ must be quite large. Vice versa, for such $c_1$ and $c_2$, the peaking condition will hold and hence we will have $\alpha + p_6\approx1$. Such conditions on $c_1$ and $c_2$ ($c_2\approx 36 c_1$ with $c_2$ quite large) apparently hold in the in fluid mechanics settings of interest.
As an illustration, here is the discrete plot $\{(n,p_n)\colon n\in\{3,\dots,10\}\}$ for $c_1=10$ and $c_2=36\times10$:
In this case, $p_6=0.72368\dots$ and $\alpha + p_6=1.0149\dots$.
Concerning the second (approximate) identity,
\begin{equation}
\alpha p_6^2 \approx 1 / 2 \pi, \tag{2}
\end{equation}
note that, if $h$ sharply peaks at $6$, then $h$ is approximately quadratic near $6$, and hence (cf. Carlo Beenakker's answer)
\begin{equation}
p_n\approx e^{-(n-6)^2/(2s^2)}/S(s) \tag{!}
\end{equation}
for some real $s>0$, where
$$S(s):=\sum_{n=-\infty}^\infty e^{-(n-6)^2/(2s^2)}=\sum_{n=-\infty}^\infty e^{-n^2/(2s^2)}
=\vartheta _3\left(0,e^{-1/(2 s^2)}\right)$$
and $\vartheta$ is the elliptic theta function.
Next,
$$\alpha\approx S_2(s)/S(s),$$
where
$$S_2(s):=\sum_{n=-\infty}^\infty(n-6)^2e^{-(n-6)^2/(2s^2)}
=\sum_{n=-\infty}^\infty n^2e^{-n^2/(2s^2)}=s^3 S'(s).$$
So, the approximate identity (2) can be rewritten as
$$R(s):=2\pi S_2(s)/S(s)^3\approx1$$
and
would of course follow from the exact identity
\begin{equation}
R(s)=1, \tag{*}
\end{equation}
which seems to be true at least for $s\ge1$, in view of the graphs $\{(s,R(s))\colon 1/10<s<10\}$ (below) and $\{(s,1-R(s))\colon 1/10<s<10\}$ (further below):
In turn, (*) would follow from the exact identity
\begin{equation}
S(s)=s\sqrt{2\pi}, \tag{**}
\end{equation}
which seems to be true at least for $s\ge1.5$, in view of the graph $\{(s,\frac{S(s)}{s\sqrt{2\pi}}-1)\colon 1/10<s<10\}$ below:
However, at this point I am not familiar enough with the theta function to prove identity (*) or (**).
Update: As was pointed out to me (and of which I should have thought myself) -- despite the dramatic graphs, neither one of the identities (*) and (**) can hold exactly for all $s$ in any (say) nonempty open interval $I\subset(0,\infty)$. Indeed, otherwise they would hold for all $s\in(0,\infty)$, because the function $S$ can be analytically continued to the angle domain $\{z\in\mathbb C\colon\Re(z^2)>0\}$. However, for $s\downarrow0$ we have $S(s)\to1\ne s\sqrt{2\pi}$ and $S_2(s)\to0$, whence $R(s)\to0\ne1$.
Yet, by the Poisson summation formula, for $s\to\infty$,
\begin{equation}
S(s)=s\sqrt{2\pi}\sum_{k=-\infty}^\infty e^{-2k^2 \pi^2 s^2}\sim s\sqrt{2\pi}
\end{equation}
and
\begin{equation}
S_2(s)=s^3\,\sqrt{2\pi}\sum_{k=-\infty}^\infty e^{-2k^2 \pi^2 s^2}(1-4k^2 \pi^2 s^2)\sim s^3\,\sqrt{2\pi},
\end{equation}
whence $R(s)\sim1$ -- so that the approximate identity (2) holds for large enough $s>0$.
Next, large $s>0$ correspond to a (somewhat) low degree of peakedness of $(p_n)$, whereas the "discrete Gaussian" approximation (!) works only when $(p_n)$ is sufficiently peaked.
Thus, the approximate identity (2) holds when $(p_n)$ is peaked (at $n=6$) but not too much -- that is, when $p_6$ is large enough but not too close to $1$. The particular numbers such as $0.34$ and $0.66$ in the OP's formula (2) play no special role except to affect the closeness of the approximation.
