Now to the questions:

During a phase change, the temperature of the system stays constant. The heat gained or lost is called latent heat. To vaporize a liquid, we must supply heat. Hence vaporization is endothermic (heat enters the system). To get the heat into the system, we must set the surroundings at a hotter temperature. Otherwise the heat will not flow. The process by which the heat flows can be any one of the three modes. We can set a pot of water in a pan on an electric heater and realize conduction from the burner to the pan to the water. We can set use a hair dryer to blow hot air over the pot and realize (forced) convection from the air to the pan. Finally, we can set the pot a small distance above the glowing electric burner and realize radiation from the electric burner to the pot.

By contrast, consider the case of a day where the outside temperature is below freezing and the air is still. A duck pond in the yard is not frozen over. Clearly, it’s water temperature is above freezing. Should we jump in the duck pond to warm up? Of course not! The natural convection coefficient of stagnate water is higher than the natural convection coefficient of stagnate air (and the difference only becomes greater in flowing water versus flowing air). The water pulls heat from us by convection faster than the air does at the same temperature, and our body temperature will drop faster than when we would stay standing in the still air. Also, while the conduction coefficient of water is larger than that for air, the main heat transfer in a fluid is convection, even in stagnate cases. The heat that flows out of our body as our body temperature drops is sensible heat.

There is moist convection (e.g., storms) and dry convection (e.g., thermals). Sensible heat (and/or dynamical processes like fronts, orographic lift, or positive vorticity advection increasing with height) initiate the rising motion to start such convection. All air has moisture, it's just that in moist convection, latent heat subsequently becomes important as a byproduct of the initial rising air, and the result becomes is something wet!

If the atmosphere is sufficiently moist, and the lift is enough to reach the LCL... then within the rising air the water vapor will begin to condensate/deposit... which thereby releases its latent heat into the air... and this additional energy help the air to rise further (since it will aid it in being warmer [and thus less dense] than surrounding air).

So convection itself is a sensible process, basically warmer air moving up into cooler air.
But latent heat release from condensing rain drops (or other precipitation forming) will add extra energy to help the air convect even more, and is vital in the formation of the strong updrafts in thunderstorms.

The initial energy that convection is taking upwards... will generally be from sensible heat. This is because the only form of latent heat transfer which would add energy to the atmosphere... would be condensation/freezing/deposition. But those basically only happen in air that is cooling down... since temperature drops are the most effective way to bring air nearer to saturation quickly.
And if near-surface air undergoes cooling, it will generally be unable to rise.
There's really one time there's condensation/deposition from the near-ground air layer: towards dawn. Coming as a result of the surface air layer losing energy via conduction to the colder ground.

(Note that in meteorology, though all motion of air of different temperatures can be legitimately termed convection... we also typically use the term primarily in regards to vertical motion. Horizontal motion of air masses is called advection. But advection can certainly move energy just the same, it's just often driven by somewhat different atmospheric processes)