1. He is in the river. [He is inside the river.]

2. He is sitting on the river. [He is sitting on the surface of the water.]
3. He is sitting in the river. [He is sitting on the bottom of the water.]
4. He is eating at the river.
5. He is eating at the riverside.
--------------------------------------
Is my expalnation correct? What does #4 mean? Does #4 mean #5? Or does #4 mean he is eating at a place on a ship or boat in the river?

4 answers

I think 2 is impossible, but all the others are OK.

4 and 5 seem to mean the same thing -- that he is sitting on the bank of the river or very near the river's edge as he eats (lunch? snack? or?).
For example, on a map, if he is eating on the river, doesn't it mean that he is on a spot of the river eating?
[Corrected]
For example, on a map, if he is eating "at" the river, doesn't it mean that he is on a spot of the river eating?
To me, “at the river” means he’s next to it, maybe sitting on the bank or maybe in a folding chair next to the water’s edge.