France in the near future!

This is very late and your budget is very low for Paris, especially for a/c and elevators. Hotels usually offer breakfast for an added charge, which is usually more than you'd pay in a nearby cafe. My best suggestion would be to search on booking.com, which filters by price, neighborhood, and other variables. But when you book, try to use the hotel's own website to avoid possible problems if anything goes wrong. You might find a great last-minute deal.

Remember that your hotel doesn't have to be right by the river or in an area close to the sights you want to see. If it's near a Metro station you can get anywhere fast. Personal safety isn't really an issue in central Paris, i.e. the 20 arrondissements within the Peripherique. Pickpockets can be an issue but not with respect to finding a hotel. Good luck with your search.

If AC is a must, there are about as many Ibis hotels in Paris as there are pigeons. They're pretty much in an IKEA meets brutalist style, but are modern and generally clean with comfortable beds and the Ibis Styles sub-brand is generally breakfast included.

Though if you aren't heavy breakfast eaters, it's generally easy to go French style and grab a coffee and a pastry from the shop down the road

Is that $100 US dollars? And how many people is we? If the $100 is USD and we is 2 people, you may have to drop either the AC requirement or the location requirement. I think you should keep the AC requirement. It could break but temperatures are extremely high right now in Paris. Take a look at Hotel Perreyve on Rue Madame in the 6th. It’s not fancy but it is clean, has AC, and is in a good location. Price should be around US$115 per night. Another option is Hotel de 3 Colleges on Rue Cujas. Good luck.

You will not be able stay near points of interest in Paris this late during the height of tourist season at $100 a night not unless you stay in a hostel with bunk beds and communal showers, Some Hostels have ac elevators and always have breakfast of some kind, If hosteling is your style say so you will get recommendation

You could try Hotel Familia http://hotel-paris-familia.com/ It's a cute french style building in the Latin Quarter. It has been pretty inexpensive for us. I'm pretty sure it has AC, I generally don't pick hotels without. There is breakfast for an additional charge. We skipped the hotel breakfast and saved some money by going across the street to the bakery. (Which is where the server was going to get pastries). The staff is very nice. The rooms and bathrooms are clean though a little worn. The location is great. You can walk to the Notre Dame, Pantheon, Louvre, and there is a metro about a block away.

Luckily, August is a slow month for hotels in Paris, many offering incentives such as free breakfast. I would book directly with the hotel itself either by calling or using their website:

Hotel Marignan (5th arrondissement)
le Relais du Marais (3rd arrondissement)
Cosmos Hotel (11th arrondissement)

For two people under 100€ (probably not all under $100), these hotels have immediate availability, some include breakfast or add it a reasonable cost, and each is very well located.

What Dick said -- go on booking.com, filter for air conditioning and price range €50-€100 for the dates you want. You can see what's available, in which areas,and what each place has. Make sure you understand the cancellation policies -- I sometimes book an adequate hotel which has free cancellation until a day or two before, then keep looking for something better knowing that we at least have a place to stay when we get off the plane.