It is correct to say "On the train" when you are inside it. either one will work This is also known as being 'on board' the train.
yes
Not a direct train, you will have to change at Peterborough
Actually, the engineer controls the train's movement in & out of the station. The engineer is in control of the train movements with one exception- when a train is put together. A hostler...
Get on the train.