I stayed in www.vrbo.com/58971 and would heartily recommend it. Read all the reviews on the site.
Happy planning.

Also would like AC and elevator due to health issues (or lower level apartment with maybe 1 flight of stairs)

Have a look at www.homeaway.com where you will find many places meeting your specifications. Suggest you not limit yourself to 1-7th as being close to a metro station or bus stop is more important. Bus, BTW, is a great way to get around if mobility is an issue, as you can avoid the walking and stairs involved with using metro. Also, AC in May is usually not really necessary in Paris (and I do not like being hot.)

vrbo 58971 is ground level, no stairs. Safe, secure, and quiet on an inner courtyard. No A/C, but it would be an unusual 3rd week of May to need it in Paris.

My husband and I stayed in this apartment in May of 2011 and really liked it:

http://www.myparisvisit.com/rental-one-bedroom-apartment-louvre-paris-59.html

The building had an elevator, but no AC. I agree with grandmere that AC probably won't be needed in May. The agency will provide you with fans if needed.

We rented again this past August from the My Paris Visit, apartment 111. Check out their website, there are lots of one bedroom apartments available for around 800 euro/week.

I loved the apartment I stayed in the 14th arr.in October. It is on the edge of the 6th so you can walk to the Luxembourg in under 10 minutes.

Here's the site: http://www.vrbo.com/209147

Read the comments from people who have stayed there. It's a fantastic location and a great price. Very comfortable for two people.

Some places that have a/c won't let you use in May anyways, I say get a place with rooms that face a quiet street so you can leave window open instead without noise issues.

May is actually a busy month in Paris, and aparmtents do go quickly, so if you have to don't discount a hotel ,there are lots in your budget that would be good value and central.

Vacation in Paris is one of the most convenient agencies to use. Do they have availability beyond 1-7? If so, you may want to consider. Many properties located in the double digit neighborhoods are only steps from the single digit neighborhoods.

There are many advantages to not staying in Tourist Central (where everything, everything, is far more pricey).

See also Guest Apartment, RentalFrance, Paris Vacation Apartments, adrianleeds.com/parler-paris-apartments.

As noted, it's very late to be looking for this May, particularly with the often recommended resources.

Beware of properties on higher floors, even if there is an elevator, as the elevators in older buildings often go out of order.

You really don't need to limit yourself to properties with a/c. I'm one who appreciates a/c, but it's highly doubtful you'll need/want it the third week of May. There's always the possibility of a spell of warmer weather, but not overnight. Many hotels won't have it turned on yet...and apartments generally have good cross ventilation.

Not insisting on a/c will increase options available.

Thanks everyone, will check out all suggested. And yes I now realize we don't need AC that will open up a lot more options. I was originally planning an august trip so I didn't adjust for the cooler month of May.