Classic asset price model in the continuous-time limit using a Wiener process notation can be written as
dS_t=\mu S_tdt+\sigma S_t dX
where $S_t$ is the stock price (not the stock return) and $dX$ is an independent random variable with normal distribution.
If we eliminate the drift ($\mu = 0$) and only focus on randomness as asked in your question we ...
For the first question, it is the standard assumption to make for stock returns if no other information is given. That's not to say it's a great assumption, but there it is clearly the only one that can be justified in this context.
For the second part, independence of returns tells you that investment for T years has cumulative variance $T \sigma^2$ (when ...