[1] You can see the Eiffel tower any time. When will you be in Paris? Right now, sunrise is at 6.30, so it will be dark at 5 am. They may turn the tower lights off around 1 or 2 am, so you won't see much of anything. If you want to see it lit up (and twinkling on the hour), plan for 10 pm or later. Right now sunset is after 9, so it won't be lit up until close to 10. In a month sunset will be close to 10, so the lights won't be on until 10.30 or later.
[2] Paris is full of great museums, large and small. Since you're in the Marais, go to the Carnavalet, interesting, partly art, beautiful building, free. The Picasso Museum is there too - only go if you are a fan. The Louvre has a lot, ancient Persian, Babylonian, Egyptian, Greek, Roman galleries, not just painting. For 19th c. painting, go to the Orangerie, it's small with a good collection.
[3-4] Paris Walks has 2 guided walking tours through the Marais, which has interesting architecture, and lots of walks in other parts of the city. The Cité de l'Architecture et du Patrimoine is a museum of architecture located in the Palais de Chaillot (which is an excellent place for views of the Eiffel Tower, BTW). Seine cruises with Les Vedettes du Pont Neuf pass some of the most beautiful buildings and bridges in the city. Do it early in the day and again after dark.
