analog-to-discrete-to-analog system sampling problem
Your first step is correct. However, a signal which is discrete in time domain does not have a discrete frequency domain representation per se, as in theory the Discrete-Time Fourier Transform may be calculated. The signal spectrum would be discrete in combination with the Discrete Fourier Transform (DFT) only if a finite representation was required.

The main consequence of time-domain discretisation is periodicity in the frequency domain. Your solution for exercise 2 should be, as well. For this, you may need to check whether the spectrum will overlap at $\frac{F_s}{2}$.

Your assumption about the discrete filter transfer function being periodic in the frequency domain is correct. Since the filter is applied in frequency domain, you should multiply it with the signal spectrum to get the output spectrum. Should the filter be smaller than the signal, the signal components outside the passband are attenuated (or set to zero, as this filter is ideal). The computation looks like this:

$$y_d[n]=x_d[n] * h_d[n] = \mathcal{F}_\text{DTFT}^{-1}\{X(e^{j\Omega}) \cdot H(e^{j\Omega})\}$$