In a reverse osmosis system with a semipermeable membrane, where the concentration of solutes is higher on one side, in which direction will there be a net movement of water molecules?

The net movement of water molecules will be from the side with lower solute concentration to the side with higher solute concentration, driven by osmosis. In the given scenario, this would be from right to left.

Source: Standard reverse osmosis system design principles

How well do dissolved gases pass through a perfect semipermeable membrane, given their uncharged and very small nature?

Dissolved gases will pass through a perfect semipermeable membrane essentially 100%, due to their small molecular size and lack of charge, allowing them to diffuse freely through the membrane.

Source: Standard reverse osmosis system design principles