Sorry, but this answer is actually the wrong one.

1. The fan does move the air. The blades are an aerofoil shape, pulling the air in front of it in. Much like a normal propellor on a GA plane, where the propellors create a pressure difference behind itself and push the aircraft along. turbofans do exactly the same, but just with an enclosure around it
2. Much of the air (upto 80% on some designs) goes around the compressor stages.The fan compresses all of the air (due to it's aerofoil shape pulling air in) evenly. It's only the air that goes through the core that is compressed in relation to the air around it.
3. The air is not heated at all at this stage, and the bypass air temp is slightly raised by the slight compression at the fan stage, and again when it passes over the combustion chamber. The heat of the air comes from it's compression in the core.
4. The exhaust velocity of the bypass air is directly controlled by how much thrust is presented to the compressor at the rear of the turbine. Say there are 3 shafts. One shaft connects the fan at the front to the low pressure turbine at the rear of the engine. The faster that the exhaust gasses from the combustion chamber spin this low pressure turbine, the faster the fan at the front spins. The other 2 shafts drive 1st/2nd stage turbines to compress some of the bypass air to direct it to the combustion chamber core

​

No doubt someone else will correct me where I am wrong, but by and large, that's the gist of how high bypass turbofan engines work