Can you help me understand this (old) school question related to RSA usage?
As in the other questions: Such low numbers can lead to curious-and-misleading effects. For example in this case: you choose $q=5$, and it happens that this is equal to $\lambda(15) + 1 = 5$. And that is a coincidence of this example, not a general relation.

Now… could you have chosen 5 for e? And follow-up question: Why not? The solution provided has

no

and

because 5 has a common divisior with n


Which question ans answer are you quoting here? Because it's not the one you linked to.
And I would actually say, there is no reason that $e$ has to be coprime to $N$ - except that it would be really terrible if anyone just tried to calculate $gcd(e,N)$. However, the encryption and decryption still work in the sense that you can retrieve the plaintexts (injective function).
Regarding the coincidence in th example: For every modulus, you have that $m^{\lambda(N)} = 1 \mod N$ with Carmichael function $\lambda$, and in turn $m^{\lambda(N) + 1} = m \mod N$.
And then, for two different primes $p,q$ this can be calculated as $\lambda(pq) = lcm(p-1,q-1)$. Alternatively, this is $\lambda(N) = (p-1)(q-1) / g$ for $g = ggt(p-1,q-1)$.
From that, we can see that if $\lambda(N) + 1 = q$, then we know immediately $p-1$ is a divisor of $q-1$.

So the mathematics &quot;work&quot;, no information is messed up, but the desired encryption is not reached at all. Is there a reason why 5 is kind of &quot;neutral exponent&quot;? Does such a thing also happen for &quot;real&quot; rsa values?

Yes, it does, but not for either $p$ or $q$. It happens for $k \lambda(N)+1$ for any arbitrary integer $k$.

In this example e=5 is equal to q, is that what makes the encryption become an &quot;Involution Mathematics&quot; (wikipedia)?

Since this is taken from the linked question - this is something similar, but not the same. An involution happens whenever $e=d \mod \lambda(N)$, because then you have that $ed = e^2 = 1 \mod \lambda(N)$. And again, it's pure coincidence in this number example, that it is equal to $q$. That is wrong in general.