I can't say anything about the Eurostar because I have never used it.
About staying in Paris overnight, I sugest you google hotels in Paris and reserve ahead. I did that and got a very nice (but small) room, about a half hour walk from the Eiffel Tower - still very much downtown! I forget the hotel's name at the moment but there are many choices, just take an hour or so to do your reasearch and compare prices. You can get good deals even in the middle of the summer (which is prime-time season for tourism in Paris).
I also suggest You get a map of the subway system and use it. It is fast, it can take You anywhere and it is also rather clean and safe. Taking taxis ran me close to two-hundred Euros in my first twenty four hours there ... not ideal.
Don't forget your camera - Paris is gorgeous! Enjoy your trip!
P.S. Walking is fun in Paris because at every turn (at least downtown and its surrounding areas) there is something to see. I put holes in my dress shoes by the end of my trip, from so much walking. I loved it there!