Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen' A classical introduction. See Cox's Primes of the form $x^2+ny^2$ for the remarks on sums of squares.

Among these primes, is there a (reasonable?) way of evaluating the relative density of those $p$ such that $2$ is not a cubic residue?

Note that $2$ is a cubic residue iff $p$ has the form $x^2+27 y^2$, but this seems rather foreign to the question of how these primes are distributed among those of the form $3k+1$. And I do not see how to apply arguments akin to Chebotarev's density theorem to deal with this problem.

There are some related questions I guess I should ask now as well. One is the same problem, but with $3$ instead of $2$. In the same spirit as before, $3$ is a cubic residue iff $4p$ has the form $x^2+243 y^2$. I imagine the problem for $2$ is easier, but expect any technique that solves one should solve the other.

A problem that I expect is harder is how to count the relative density of those $p$ such that the equation $2=3x^3$ has no solutions modulo $p$. But really, even pointers on how to say anything meaningful about these $p$ are welcome. The difference with the previous questions is that asking this is the same as asking that $k+1$ is not a cube modulo the prime $3k+1$. I have found harder to see what to do in this case.

Finally, if it turns out that these densities are not zero, how does one go about finding the density of those $p$ that satisfy all these conditions simultaneously?

(Ideally, the techniques lift to other situations, such as studying fifth powers modulo primes $p=5k+1$, etc, but even methods exclusive to the case of cubes are very welcome.)