Conjecture 1 does not hold as for any even $n$, (2) has a solution:
$$(x,y,z)=(2,\frac23(2^n-1),2(2^n-1))$$

Likewise, Conjecture 2 fails as for $p=2$ and any even $n$, (1) has a solution:
$$(x,y,z)=(4,\frac2{15}(4^n-1),\frac25(4^n-1))$$