Does it matter what electrolyte we use for a Galvanic Cell?
The electrolyte is very important for the functioning of the galvanic cell. The emf that the cell generates depends on the type and the concentration of the electrolyte used. In the Daniell cell, the reaction would be:

$$\ce{Zn + CuSO4 -&gt; ZnSO4 + Cu}$$

As per the Nernst's equation, you get the emf of the cell to be:

$$\text{E}=\text{E}^\text{o}-\frac{2.303\text{RT}}{\text{nF}}\log_{10}{\left(\frac{\ce{[Zn^{2+}]}}{\ce{[Cu^{2+}]}}\right)}$$

Clearly, the emf of the cell depends on the concentrations of the cell. Dipping the rods in $\ce{NaCl}$ would not make the cell work.

And for why the salt bridge is needed, think about why you would build a cell in the first place. To generate power in a usable way right? If you let the solutions mix, they would react directly, and the energy would be released as heat instead, and not as electrical power. You want the ions to move towards the electrodes and get discharged there.