Hi @awagner
To answer your question exhaust fans will work pumping heat out but air must be pulled in from outside some where else in the house and that could be hot air from outside. But if you clever you can install air vent on the cool side of the house let cooler air come in. Some smart houses pull air from under the house or a basement where temperature is cooler and near constant.
You could use Air con is best as it is a closed system in your house many wall units are very economical these days if used at peak hot time and not 24/7, make sure it uses inverter technology as this saves power use.
Also a dc powered ceiling fan can be told to spin air up or downwards this left on all day will mix the air in the room and the room will feel cooler ie no hot roof air layer. They are very economical in power use especially on low speed. I find hot air rises to the roof heating the roof materials and even after you turn a fan on the roof remains heated and quite warm for a few hours the trick is to stop roof materials warming up.
I have bathrooms with tile from floor to ceiling that acts as hot or cold thermal storage in summer the cold tiles overnight stay cool until late in the day leaving the 2 bathroom doors open helps cool indoor air, in winter we close the doors. I should consider heating the tiles with sunlight some how then winter time open the house to warm air. This works not to a huge degree but enough to take some heat out of the house as colder air rolls across floor then mixes with upper air via fan.
Another way to start thinking about solving your problem is find ways to stop the heat going in that way you may not need a lot of power to cool your home.
Plant bushes and or trees or pot plants on wheels on the hot side of your house. Things like a grape vine on a pergola type structure works well as summer the leaves are very cool and shady and winter all the leafs fall off for winter sun warming of your house. My first grape vine i grew on wires across my house side and the windows that was a huge heat reduction.
Put solar film on your windows and doors.
Use blinds to stop the heat. I am looking at electric blinds I can automate for different season changes (Just my pipe dream still)
Use a pelmet over the window as this stops heat and cold on the outside of the blind from starting to form a convection current in your room.
Some security shutters are more insulated than others by personally i think they over rate at insulation still only block 15% in my opinion and window still can be warm as they are made from aluminium and the sun heats them and between the shutter and window convection and some secondary radient heat still hits the windows and they feel warm mid summer.
Another couple of rooms I sewed on a silver lining like a flexible silver polyester that worked well. They use light weight mylar in space craft hulls thats where i got that idea combine with a pelmet is 80% heat block out. Maybe we could get NASA showing us a couple projects in here lol
Well down stairs the only other way is sun hitting you bricks and heating them by the end of a hot day. All you can do is shade walls I don't know if expanding foam would work between gyprock and outer bricks or double brick I would hire a company to do that any how if such a thing exists.
Solving heat problems is tricky and may take a few steps I just wish they build all houses to new standards so we don't have to worry about any of this.
Project time line could be
Air con or ceiling fan or both I use fans mostly and air-con on super hot nights only., greenery outside and window foil ( the silver or gold reflecting type is best think astronauts gold helmet visor it works well.)
