I have an old house and I have no air conditioning, and in fact do not miss it at all. I go along with general conversation about air conditioning, but secretly I think people who have it are complainypants wusses. :)
I have never noticed any mold problems, and I live in an area that gets very hot and muggy during the summer.
Admittedly, old houses (mine is nearly 100 years old) are built to handle hot weather, and modern houses aren't always designed with the cross-drafts etc. But here's how to maximize the coolness:
Keep the windows shut during the day, and the lighting low -- curtains drawn if you aren't using the room or need the light. In the evening, when it's cooler, you open the curtains and the windows (and solid exterior doors, if you have screen doors) wide open. Then the cool air comes in.
Ceiling fans are helpful if you have them. Do some internet research to make sure they're revolving the right way (it's opposite for summer and winter, depending on whether you want the warm air to go up or down).
When you have windows open, cross-drafts are what you want, from one side of the house to the other, or within the same room.
Our doors do stick in the summer. We just don't close the dining room door all the way during the summer. In the fall it will return to its former size without a problem. If the swelling is really inconvenient, you can shave a bit off the door. I guess I hadn't even realized that this is no longer normal.
If it's really hot in the afternoon, this is what siestas are for! Also porches and hammocks!
If it's really hot, you can put your T-shirt in the freezer for a bit before you put it on. Nice! At night, put your pillow in the freezer for a bit before you go to bed. Also nice! And lemonade with ice cubes is your friend generally. And iced tea.
I have never lived in an air-conditioned house. When I visit them I find they're clammy and sterile, with all that chilly unmoving air. It might as well not be summer. Love a breeze in a house! The studies do find that air conditioning trains people to be less adaptible to changes in temperature. Come on out and join the non-wussypants un-air-conditioned brigade.