You are correct about the routing of air in a high bypass turbofan. The reason that the bypass air produces so much thrust in a high bypass turbofan is because of the greater volume of air that bypasses the core. In a 4:1 bypass turbofan you have 80% air bypassing and 20% entering the core. The core airflow is responsible for about 30% thrust while the bypass produces about 70%. The reason the core produce a higher proportion of thrust compared to its airflow is because of the heat energy and added massflow from combustion.
The identification of the N1,
N2, and N3 sections or spools is determined from the front of the engine to the back ie. the fan is N1, the intermediate compressor section is
N2, and the high pressure compressor section is considered N3, that is if it has N3. The turbines are reverse order since the sections or spools are on concentric shafts the turbines driving N3 are first, then
N2, and then N3.