Paris, 3 days - no car.

You should not have a hard time finding a room in that range unless you need a big American-style hotel, a top luxury hotel, or have to get 4 people in a room. You could walk to many places from the 7th arrondissement, but you don't have to walk everywhere - there are buses and the metro which are great ways to get around. At the end of August, you may well want to find a place with air conditioning. You might start with Rick Steve's suggested hotels near rue cler. They are very convenient.

I assume the train station that head to Lyon is Gare du Lyon which is on the opposite side of town from the Eiffel Tower which is at the far western edge of Paris. Paris is very walkable but these places are not within easy walking distance. I'd get a hotel that is central and near the river and plan to use the metro to get about. There are half a dozen train stations in Paris, each of which serves a different part of the continent. We like Henri IV Rive Gauche which is near Notre Dame and a nice walk through the Tuilleries up the Champs to the Arc de Triomphe. It is nice to be near the river as strolling there in the evening is quite lovely. But for that money, you will find good places to stay all over town. I agree AC is a good idea that time of year.

Last summer I enjoyed Hotel Duquesne***, located in the 7th near the Ecole Militaire metro stop. It's in your price range.

Hotel Relais Bosquet, walking distance to the Eiffel Tower, Champ du Mars, around the corner from the Rue Cler with its market stalls, cafes and bistros. Metro stop is Ecole Militaire. Lovely hotel, very comfortable, good breakfast choices, WiFi works great. I'll be returning there for my second stay in October.

Rick's most recent PBS special on European travel basics had a few snippets filmed there.

Have you read Rick's hotel section? He has lots of options in all parts of town.

Gosh I love these forums! Thank you all and will check out all three of the hotels mentioned so far as well as Rick Steve's section on hotels - thanks again, Phred! (I didn't realize there was one... I started with the forums and haven't explored the site that much..) It is so nice to get personal recommendations.

You might also get on www.tripadvisor.com, Paris forum, for specific tips on just Paris. You can fill three days and you won't ever need a car in Paris. Paris is imminently walkable and the metro system is easy to figure out with a couple of carnets (10 packs) of metro tickets. Check out hotels in St. Germain or on the islands, both with handy metro stops, although you can walk to most things from the 4-5th Arr. Be sure and get air conditioned rooms in August.

No Car in Paris is actually a very very very GOOD thing. Jump on the metro and go wherever you need to go because it's a vast system with many destinations. If you're uncomfortable with the Metro then just hail a taxi. They're not "that" expensive.

Nowhere is within walking distance of all the places you mentioned, and do not try to drive your own car in Paris. The public transport network in Paris is mostly very good. Learn how to use the Metro, but don't neglect buses - some of the routes are quite scenic although they can be delayed by traffic.

Two sites to check out.

parisbytrain.com

tomsguidetoparis.com

The Metro brings you for your 3 days anywhere in Paris, to my opinion learn to use it asap. If you use Metro Line 1 most famous sites are within (easy) walking distance and also stops in Gare de Lyon. From Arc de Triomphe (station Charles de Gaulle Etoile) Line 6 goes to Trocadéro to visit the Eiffel Tour, in opposite direction Line 2 has a stop in station Anvers for going to Montmartre.

However, don't spend all your time in the Metro because you won't see anything. Mix in some sightseeing buses or bateux (sp?) and lots of walking.

A fit walker could hike all the way across central Paris in a couple of hours. Using the mass transit system to reach the areas where you want to stroll is easier, especially when the weather turns against you. Its website has lots of tour information and will plan routes for you, but it is worth practicing before you leave so you will feel comfortable once on the ground: http://www.ratp.fr/en/ratp/c_21879/visiting-paris/
Parisbytrain helps understand the system. I look for hotels that are near a Metro station, especially if it has a couple of lines to choose from. Above ground, the bus network is extensive and there are some streetcars too The frequent poster here named Kerouac has written (and photographed) many neighbourhoods visited either on foot or bicycle.

Last fall my wife and I got a Apartment (airbnb) on Rue AMeile in the 7th a few blocks east of Rue Cler. It was great location.
a nice walk to the Eiffel tower, Short walks to Napoléon's tomb, Rodin Museum and several others close. If your up for a little hike you can walk to the Orsay and Orangerie. The bus stop at Ecole Militare has a bus that will take you to the arc de triumph take a stroll down the Champes Elesse catch the bus back to Ecole Militare stop. The La Tour Marburg Metro stop is just 3-4 blocks east of the location we stayed at.... Most places don't open until after 9 am, since we are early risers we made a big breakfast in the morning and headed out....

Any of the arrondissements #1-8 are very handy to catch the metros & walk around. I've stayed on both sides of the river and enjoyed each experience. I generally search on expedia.com or booking.com for my price range & look for the best reviews. For August, be sure to choose one with air conditioning.

Peggy, it has been our experience that it really doesn't in which area one stays. Very easy to take the Metro to any area and then just walk. Only rarely do we even bother with cabs -- rain or perhaps going to or from train station with luggage.