When a pot of water starts boiling, does that mean that the top surface of the water was heated repeatedly by convection? Meaning that it was initially at the bottom of the pot, then it rose up to the top when heated, then circulated back down and then up again until it reached 100C? (Is that what actually happens or is it just a way to explain how boiling happens?). Also, is there another way to explain why hot water/air rises other than "it became lighter because of it's lower density"?
The idea that water heated at the bottom of a pot rises to the top and returns to the bottom by convection is correct. The convection happens because hot water has a lower density and thus rises up to the surface where it cools and sinks down to the bottom again. When water starts boiling there is the additional effect of the formation of water vapor bubbles that have a much stronger buoyancy than hot water. Thus these water vapor bubbles by their fast movement to the top where they leave the liquid also produce a strong upward water current leading to an increases convection in the pot.
To answer your first question, depending on the circumstance around the water, yes convection is what heats the pot. To your second question: Hot objects rise because of what 'heat' is. How hot object is depends on how fast the molecules/atoms which make it up are moving, with the hottest objects having the most movement and the coldest having the least (which is why we have absolute cold temperature, but not hot). So the atoms that are moving around more quickly push against others with more force, causing hot objects to expand. Biggest objects push others out of the way, but, having nowhere to go, the hot objects (air and water) rise to the top. This is a GENERAL rule with some exceptions, i.e. water, which, due to molecular structure expands when it freezes.