I haven't dealt with a travel budget for a while - but I will put on my thinking cap. It doesn't make sense to me at all to go into Paris for one day - then leave Paris - and then return for 4 days. Either spend your Paris time at the beginning - or the end. Don't split it up.
I don't know how old you are. When my husband and I were younger - we could do a red-eye from the east coast of the US - pick up a car at the airport - and then drive 2-3 hours to get to wherever we wanted to go. We'd be tired - but we'd arrive in the early afternoon - take a nap - go for drinks/dinner - and then be adjusted time-wise the next day. We are over 60 now - and can't do this anymore. When we arrive on a red-eye flight - we are ready to go to sleep for a few hours. In light of your budget - I'd avoid a red eye arrival booking first day you arrive (where you can check in early by paying for the previous night) - and go to your first destination outside Paris from CDG. You'll be really tired - but your room will be ready for you to crash for a few hours until dinner (and you can freshen up too).
I would definitely rent a car. Driving in France isn't difficult - and a car gives you a lot of flexibility. Get one with a GPS. We do that everywhere when we rent a car these days. It's well worth the few dollars extra unless you really like to buy and read maps.
I cannot imagine riding a Segway in Paris (I rode one once in a museum demo). There is simply too much vehicle traffic in the streets - and pedestrian traffic on most sidewalks. Plus - Paris is the kind of city where you want to poke around - explore the side streets and byways. At a leisurely pace. What you save on the Segway - you can spend on the GPS for the car.
All of France is "wine country". I suggest buying a copy of the Green Michelin Guide (sightseeing) - and see which areas look amusing to you. No matter where you go - you will be able to find wineries and good local wines. FWIW - I've read that Giverny is a very crowded and not that wonderful a tourist destination. Perhaps not worth the trouble. Also pick up a copy of the Red Michelin Guide (hotels and restaurants). Note that in France - as in most of Europe - you will probably pay more than you are used to paying in the US for rooms and meals because of the lack of standardized and relatively inexpensive chain hotels and restaurants you find here.
I think for a first trip I would recommend a trip to the Loire - and perhaps another area that interests you for a particular reason (like Normandy for WWII stuff - the Dordogne for great food - etc.). Look at the Green Guide for 3 star (worthy of a trip) destinations. If you are traveling in high season - between April and October - make reservations. A lot of nice places don't have many available rooms - and you don't want to show up in a small town only to be told there are no rooms available. And I would definitely recommend having at least one meal in a higher end (2 or preferably 3 star Michelin) restaurant. Outside of Paris - where the food is still wonderful - but the prices are less expensive.
If you want to brush up on your conversational French before your trip - get some Pimsleur stuff (many levels - buy the one that suits your current abilities best - some Ebay stores have the best prices).
I am not sure why people care that locals know they're tourists. I live in a Florida golf resort community. And it's always easy to tell who the tourists are. They're the ones with the Scottish accents and the bad sunburns

. On the other hand - it is always great - no matter where you go - to respect a country enough to speak at least a little of its language and learn a little of its culture and its customs. And in Paris - like many major cities in many countries - you never want to put valuables like wallets in places like pockets where they can picked. I always carry my husband's wallet in my purse - which is always slung across my arm along my front - not my back. As for backpacks - I have a rather chic one I like a lot that I bought in Japan. A place for sweaters - guide books - umbrellas - etc. If you have to carry a lot of stuff around a city - a backpack is as good a place to stow your stuff as any.
The gold ring thing is something you will undoubtedly encounter in Paris - and you just have to walk away. One problem with Americans is they're afraid to seem rude (but it is very justified in this case).
Don't be afraid to explore - and have a great trip, Robyn