No Stupid Questions

if you have one and only one opening into your room, id believe setting the fan to exhale is optimal, as the logic would be similar to radial fans in low airflow situations, where in that situation, its ideal to move hot air out.

if you have 2 windows, I think it was one intake. if you have a window and a door (to the rest if the house), placement depends if youre okay with dunping heat into the other part of the house or not.

theres a lot of factors though, as it pertains to the rooms internal pressure and the condition of the air/sunlight outside of the window, as well as how high the window is

It depends on the windows placements that are open and temperature of the building sides vs the location which can be time of day dependent. It is not always good to open all windows as it can be valuable to create an airflow that does what you want. Air pressure goes down as you go up in height so its good to have windows open lower on the cooler side and higher on the higher side. Its possible the whole heat risese effect helps this to im not sure. This should create its own airflow that is slight compared to a fan but very consistent and can help a lot on its own. From there I do think pulling out seems to work better than pushing in but it is way better if it covers the whole opening as much as possible. So like if you can use foam insulation to make a form fitting collar for the fan that sizes it to sit in the center of the window opening hole and seals it to the edges. That will work better.

Happy to hear you solved it for now. For a more long-term solution, consider investing in an air conditioner. And make sure it also has a heating mode (a.k.a a heat pump), it should cost pretty much the same and is the most efficient way to heat a home in winter.

The most efficient way is to force cold air into the space while also forcing hot air out. The best way to do this is to have maximum airflow from one end of the house to the other, often by opening all of your windows and using multiple fans to direct the air. For example, an intake fan pulling air in at the north end and an exhaust fan pushing it out at the south end.

I've tried it many ways. The most efficient method using a fan is to open the windows in a room on the leeward side of the house, then place a fan a few feet in front of the doorway, blowing air into the room. It's more efficient because the impelled air will entrain surrounding air moving a greater volume through the doorway and out all the windows in that room. Then open windows in the farthest rooms to get a cross breeze.

Also look into an attic fan. They exhaust hot air from the attic, and it makes a world of difference in keeping your interior cool. Another trick is to mount shade cloth 6 inches away from the sunny walls of your home. It absorbs most of the solar radiation and keeps your walls cooler.

Have an fan pushing out on the ceiling in the middle of the house would work good, it will get the hottest air out. But also keep in mind: the air has to come from somewhere, if the house is sealed (new windows) this will not work well.

I'm not a physicist. But I did live in an apartment at university, and shortness of funds plus hot weather meant experimenting with various box fan configurations.

What I found most optimal was to open two windows on opposite sides of the dwelling, with the one box fan aimed outward, using cardboard to block the openings around the fan. In my case, the choice of egress (ie air flowing out) window was based on: the window which best fit the box fan's shape, proximity of noise to the bedroom, and the quality of the window screen.

As for why the fan points outward, this sends the heat of the motor (60-100 W) out of the dwelling, rather than drawing it in. Also, if facing inward, the high airflow at the tips of the fan blades would tend to draw small flies into the dwelling. But if the fan is at the egress window, then the ingress airflow will average out over the full surface area of the ingress window, producing a lower peak airflow rate, akin to a gentle breeze.

If you have a multi-floor unit or house, it would be optimal to place the fan at the highest egress window, to take advantage of heat naturally rising. Opening multiple ingress windows will quickly cool those rooms, while also reducing the peak ingress airflow and resulting drafts (eg blowing papers off tables). Of course, it's necessary to open all the doors to form a path between the ingress and egress windows.