Lets start with the answer to the first question:-
(1)The Purpose of the solution in the galvanic cell is to facilitate the conversion of $M^{2+}$/$M$ or $M$/$M^{2+}$ as the case may be. Basically the solution contains Metal Ions which when required may convert to the Neutral Metal Atom. Similarly the electrode contains Neutral Metal Atoms which when required may convert into ions.
For Example, in the $Cu$ Cathode:-
$$\ce{Cu_{(aq)}^{2+} + 2e ->Cu_{(s)} }$$
Over here an atom from the aqueous phase loses two electrons and moves onto the cathode. BUT THIS WOULD NOT HAVE BEEN POSSIBLE HAD THERE NOT BEEN A SOLUTION AROUND THE CATHODE CONTAINING $Cu^{2+}$ IONS AROUND IT.
Thus we need a soultion around the electrodes.
Moreover the solution in both the halves contains both positive cations(like $Zn^{2+}$ and $Cu^{2+}$) as well as negative anions like $Cl^-$ or $SO_4^{2-}$ if the solutions are made up of $\ce{ZnSO4}$ and $\ce{CuCl2}$ .
(2) Electrons do not leave the $Zn$ atom unless and until the two electrodes are connected by a wire, since when electrons leave the metal atom they need to get used by another metal ion, which is what happens in the $Cu$ cathode
$$\ce{Cu_{(aq)}^{2+} + 2e ->Cu_{(s)} }$$.
Moreover current(electrons) does not flow in a broken circuit thus the two electrodes need to be connected by a wire, and the two solutions need to be in contact too, possibly through a salt bridge or a permeable membrane.
