A basic question regarding frequency analysis of an EEG signal
The first $5000$ points (well, $5001$ to be exact) contain all frequency information, because the signal is real-valued and, consequently, the spectrum is conjugate symmetric:
$$X[k]=X^*[N-k]\tag{1}$$
where $N$ is the DFT length. But from $(1)$ we cannot infer all values of $X[k]$ from any arbitrary collection of $N/2$ (consecutive) points of $X[k]$. So you're right.

You got that right.

This is a bit tricky, and you'll need Eq. $(1)$. Think about which values of $X[k]$ you can derive from the index range $k\in[8001,9999]$. It should turn out that in fact you got everything you need.

They should specify what they actually mean by $\pi/2$. I would expect this to be a normalized frequency in radians, i.e., $2\pi f/f_s$, where $f_s$ is the sampling frequency. If that is the case, then $\pi/2$ is just half the Nyquist frequency. Draw your own conclusions.