They first one is technically correct, but it is not natural in U.S. English. You could use "near" instead. The third one is not accurate, since the Eiffel Tower is set back from the riverbank. Again, you could use "near." The second one is good. And, since the Seine is widely known, it would be typical to omit "river."

The Seine River, the River Seine, and the Seine are all correct. In North American English it is common to put the word "River" after the name. In the UK it is common to put the word "River" before the name.

All are grammatically and semantically correct. However, it may not be a fact. "next" is very close to the river. "by" is a further away from the river but still not considered to be far. "on" means it is on top of the river! I know that this is not the case. I have never been to Paris and do not know how far or close to the river. The Seine river. The River Seine. The Seine is world-famous enough to omit "river".