Regarding the Galois group of the factor $g(x)=g(q,x)$of these Fekete polynomials that is conjectured to be irreducible, here is PARI code that counts, for each prime $q \equiv 1 \bmod 4$, $17 \leq q \leq 41$, the number of primes $p$ for which $g(q,x)$ modulo $p$, has $n$ linear factors, for primes $p$ up to half a million, stored in the matrix $c[q,n]$, done for $2\leq n\leq 12$.
for(n=2,12,forprime(q=17,41,if(q%4==1,forprime(p=2,500000,if(matsize(polrootsmod(g(q,x),p))==[n,1], c[q,n]++)))))
If I did not make any mistakes, then, for these four primes $q=17,29,37,41$, here is the $4$ by $12$ matrix showing $n=1$ to $12$.
[0 12634 0 3084 0 538 0 72 0 3 0 0]
[0 12486 0 3101 0 530 0 50 0 9 0 0]
[0 12637 0 3187 0 519 0 59 0 2 0 0]
[0 12519 1 3081 0 543 0 65 0 6 0 1]
What appears to happen is that $n=0$ which is not shown here and which I only tried for $q=17$ gives about 60%. Please let me know if you find any mistakes, thanks!