nt.number theory - additive structure in a small multiplicative group of a finite field?

Probably not, assuming $p$ is fixed and $n$ is large enough.
Have a look at section 5 in my paper A49 in: http://www.integers-ejcnt.org/vol7.html (for some reason the journal doesn't allow direct links to papers although is free access).

In the notation there, let $R(x)=x^{n-1}+1$. Note that, as a
consequence of your hypothesis 3), $beta^{n-1}+1 in <beta>$, which
implies that the order of $beta^{n-1}+1$ is at most that of $beta$.
This will give an upper bound for $N$ in terms of $n$, using your
hypothesis 2). I haven't done the
calculation, so I don't know if this upper bound contradicts your hypothesis 1). Note
that the bounds that I get are probably much weaker than the truth, see e.g.,
the conjecture of Poonen's discussed in the paper.

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