The sum of independent interarrival times for the poisson process is a gamma random variable. in general does the sum of exponentials have to be independent to sum to gamma?
also this would produce the arrival time for the nth event. is this arrival time independent of other arriuval times? e.g. S4 independent of S9 where S denotes the arrival time at 4 & 9 respectively.
To answer your first question, take a random variable X that is an exponential. Now consider $Y = 6X$, which is a sum of six non-independent exponential random variables. Is that a Gamma?
To answer your second question, compare $P(S_2 < 1 \mid S_1 > 5)$ and $P(S_2 < 1)$. If you instead ask about the time between arrivals, that is independent for a homogeneous Poisson process.