Paris is a wonderful city I never tire of. Here are a few of my suggestions:
1.) The fastest, cheapest way to get to the city center is the Metro which means you don't have to worry about traffic causing any delays, which is helpful for your return. Taxi from CDG is supposed to be a flat rate of 55 euros, but it may vary depending on luggage and drop off location. Metro is about 12 euros each from or to CDG.
2.)Need more information about where you will be staying, what your budget for dinner is, and what you consider romantic. I have been dying to try Les Papilles, a bistro in a wine shop on the Left Bank. No menu, fixed price, must book in advance. Les Foodies, in the Marais, is a sweet fusion bistro, the chef is half Italian (mother) and the other half Thai. Delicious food you won't see anywhere else. Le Petit Retro near Place Victor Hugo is a great option too - a quintessential French Bistro down to the tiled walls. A little further down the block (Rue Mesnil) is a sweet little bistro called La Coincidence, totally a local favorite, but English is spoken.(all these places).
3.)Buy metro tickets at the metro station. To save money you can buy a "book" of ten - the machine just spits out ten tickets. If you were staying a little longer the Navigo Decouverte is a great option. It runs from Monday to Monday, unlimited use of the system with a card. But if you are there less time it pays to just buy a book.
4.)Metro tickets you can buy at any metro. Museums you can buy online if you know you will visit.
5.) I would research or google jazz in Paris by the month. Lot's of articles available, Time-Out Paris, etc.
Another of my favorite restaurants is near the Place des Vosges called L'Ange 20. Father in the kitchen, son front of the house. Unpretentious bistro, classic French but not heavy, very friendly. Near the Picasso Museum is a great little Breton crepe place called Breizh Cafe, savory and sweet crepes, high quality. If you are anywhere near the Canal St. Martin (very cool bohemian vibe, lots of ethnic restaurants) you should make a beeline to a boulangerie called Du Pain et Des Idees. There is always a line (almost always) for their fabulous wood fired oven bread, and especially their "escargot" which is croissant dough but prepared like a jelly roll. My favorite is the pistachio creme with chocolate. Order a cafe creme, and sit outside on the communal picinic tables. Folks drive from all over to go to this award winning bakery. Mostly give yourself time to wander all over Paris. The museums are fine, but just getting lost in the atmosphere is the best thing about Paris. One more tidbit. If you are on Ile de la Cite checking out stuff, just over the bridge is a tiny little chinese place with very good soup dumplings for almost nothing, great for a quick cheap tasty lunch, called Jixaou's Buns.
Have fun!!