Question

1. The ship is moving on the river.
2. The ship is moving in the river.
[Which preposition do we have to use?]

3. He is floating down on a wide board on the river.
4. He is floating down on a wide board in the river.
[Do we have to use 'on the river' here?]

5. He is fishing on a boat on the river.
6. He is fishing on a boat in the river.
[Do we have to put 'in' or 'on' before the river?]

Answer 1

1 and 2 both are OK.

3 makes more sense than 4 does. 4 seems to indicate that he is under the surface of the water, and that would not be good. Actually, 3 and 4 would be better phrased like this: He is floating down the river on a wide board.

5 and 6 seem to mean about the same thing.