Paris Apartment Recommendation?

Check to make sure any apartment rental is legal. Paris is cracking down on Air BnB and others. It is rare to find a/c in a rental property in Europe.
We have used Paris Perfect and Just France for rentals in the past. The apartment buildings had small elevators. We paid much more than your stated budget years ago but we had two or three bedrooms for our friends or family members.

There are many threads on this forum about finding legal apartments and I’m sure you will get responses that address this issue. I can’t comment on that but I can tell you we rented listing #6583168 on VRBO/HomeAway last May. It is located on Boulevard de Sebastopol about two blocks from the Pompidou Centre near the Marais. It’s close to Metro Les Halles. Monoprix for groceries about two blocks away. This studio apartment was well equipped with a very comfortable queen size Murphy bed. There is an elevator to the sixth floor but you still have to navigate a few stairs. No AC but wasn’t needed in May. Equipped with fully furnished kitchen and washing machine. Host was very welcoming and helpful.

We used Vacation in Paris this September and I would highly recommend them. Our 2 bdrm apartment was in the Rue Daguerre area in the 14th Arr and a block away from 2 different Monoprix's and the market street of Rue Daguerre. Perfect setting. Metro was close and only 15 minutes or so to the top tourist sites. VIP were great to deal with, have a number of apartments across Paris, and many would be in your price range. Just sort through their search engine for the details that are important for you. Metro stops are everywhere, so you shouldn't have too many problems there.

Here's VIP's website = http://www.vacationinparis.com/index.php.

Most residential apartments do not have air-conditioning, because they don't feel the need to use it for heat waves.
I'd opt for the Citadines at Richard Lenoir, which is steps to the Bastille open air food market, and has air-conditioning and has a modern elevator, and kitchen.
Good bus connections, if you don't want to use the Metro.

"professional and timely as one expects a doctor to be" Is that a compliment or an insult? LOL, I think I have had two doctors appointments in my life for which the doctor was on time. By the way, I know someone who highly recommends the Printemps apartment at 68Rivoli.

If you want an apart'hotel in the Marais, this one is very similar to a hotel, and has a small open-air food market and supermarkets, boulangeries, etc. Good connections to Bus lines, too:
www.booking.com/hotel/fr/le-roi-de-sicile.en-gb.html