You are correct that hot air rises. Most matter (gas, liquid, and solids) expands as it becomes warmer. Therefore the material becomes lighter. Any part of the air or ocean that is hotter than the surrounding air or water will rise up because it is lighter than the colder air or water. The reason why the ocean, although warmer than air, does not continue to rise is because even though the water is warmer the water is still heavier than the air above. If the water continued to increase in temperature however it can become a gas and rise above the cooler air.

The answer to this question is all about something scientists call density. Density is just a fancy way to say how much material is in a space. Instead of material and space scientists call them mass and volume. Therefore density is the measurement of how much mass is in a given volume. If matter is less dense than the surrounding matter it will rise until it reaches other matter that is of equal or less density.

You can test this by feeling how heavy a block of wood is compared to how heavy a piece of metal is of equal size. Even thought the wood and metal are the same size the metal is a lot heavier. Thus the metal is more dense than wood. Next place both items in a bucket of water. The wood floats and metal sinks. This is because the wood is less dense than the water and metal is more dense than the water.

The ocean behaves just as air does. The warmest less dense water is on the top of the ocean. The colder, more dense, water is on the bottom. The air (or atmosphere) is the same way the warmest air is on top and the coldest air is at the bottom. So the reason why warmer ocean water does not continue to rise in the air is because the warm water is more dense than the cooler air. Now again if the water continues to heat until it becomes steam it will rise up into the air.

1) Air temperature is a function of air pressure.At low elevations, air pressure is really high, because there is a LOT of atmosphere above you. At high elevation (on a mountaintop), there is less atmosphere above you, so the air pressure is lower. This generally causes low areas (like in Death Valley, 282 feet below sea level) to become very hot, while nearby mountains (like Telescope Peak, 11,000 feet above Death Valley) will be much cooler.

2) During the daytime, when the sunlight hits the land surface, it is absorbed, causing the surface to heat up (think about how hot pavement gets on a sunny, summer day). The hot surface then heats up the air near the surface, causing it to rise (as you said in your question). However, as the air rises, it cools (because pressure decreases as you climb through the atmosphere). Eventually the air cools enough so that it sinks back down toward the ground. This process happens everywhere on Earth, so, on average, air temperature decreases with increasing elevation. This actually happens at a known rate, which scientists call the "lapse rate". For most places in the western US, the temperature drops by ~3.5 degrees with every 1000 foot gain in elevation.

The sea level is warmer, because hot water rises! It would be fun to do an experiment with hot water and food color in a glass, and then slowly add cold water, and see whether the cold water seems to sink or go to the top of the glass. Maybe I'll try it. Maybe you'll want to try it too.