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
Get on the train.
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...
Not a direct train, you will have to change at Peterborough