Great questions!

In short, the temperature of the ocean water must be lower than the temperature of the air moving over it for a fog bank to occur.

In other words, the water and the air can be any temperature, as long as the water is significantly cooler than the air.

To transport warmer air over cooler ocean water we need wind. When winds are blowing in the right direction to bring warmer air over cooler ocean water, fog will start to form. They will start to disappear when the difference in temperature between the water and the air decreases. This requires either the ocean surface to get warmer, or the air to get colder.

We often see fog banks form along California’s coast; it is because the ocean water along our coast is very cold compared to the rest of the ocean. This is why the fog banks that form at our coasts are quite narrow – if you traveled in a boat out to sea you would soon move past the fog bank into clear skies. This is because during the summer months, cold water up wells to the surface from the bottom of the ocean along a narrow strip of coastline. At this time of the year the wind usually blows towards the California coast from west to east, bringing with it a lot of moisture that has evaporated from other parts of the Pacific Ocean. When this air approaches the cold water at the California coast, the moisture in it condenses into visible droplets, creating fog. Since this wind is blowing from the ocean towards the land, the fog is sometimes blown onto the beach or even further inland, until the sun burns it off.

This type of fog formation is sometimes called ‘advection fog’:

See this picture here