I am a computer programmer, this is exactly how it works. Why else do you think electronics have fans in them if not to blow fresh, crisp wifi in and stale, soggy wifi out?
Putting a fan begin your router won't boost the range because photons emitted by the router's antenna won't be affected by moving air from the fan. Putting a floodlight however...
Alright, so according to Bernoulli's principle says that moving fluids result in a lower air pressure. Light and all electromagnetic waves are fastest in a vacuum. Lower air pressure is closer to a vacuum. So... Marginally? I have no idea how much but I'm guessing it's miniscule enough to need special equipment to detect. Not worth it. Plus the fan itself could block the waves. The fields around the wires powering the fans would have an effect as well. All of this is going to be super minor but I think the physical blockage of the fan is going to have more of an effect (but still teeny tiny) than anything.
If it has metal blades then it will reflect some of the radio signals, making the transceiver more directional. With how it's set up in the post, it could potentially be a benefit to devices that face the front of the router and fan, but a disadvantage to devices behind the fan. Same logic with that Facebook trick of putting tin foil or cut up drink cans behind the antennae.
However, most newer and higher end routers use beam forming antenna arrays which are already directional and can automatically focus the signal toward your devices. Having reflectors around those can actually interfere with the antenna array and decrease speeds for all devices.
Technically yes, but in practice any gains are going to be counteracted if not outweighed by the electromagnetic noise from the fan's motor. To avoid that interference and see any real improvement in your signal strength, you'd have to either use a fan with a shielded motor (the last such model went out of production in 1953, so good luck finding one) or a fan driven by an alternative power source such as a water wheel.
The wifi beams come out in all direction. You can help boost the wifi by placing a mirror behind the router. Then the rays will be reflected back to you and not wasted.
Bernoulli's principle says moving air has lower pressure and em waves travel faster in low pressure air. The effect is going to be incredibly minor and orders of magnitude less of an effect than stuff like walls in the way or other wifi routers interfering.
I'm pretty sure thats unpossible, try a microwave and some garlic it should boost the signal, direct the antennas from the router to the most metallic part of the microwave and the microtextural bogoconductor + full-duplex planck magnetoms as well as the quantized garlic aroma should distribute the signal more evenly around the room (even through walls) than the antennas ever could. Trust me I'm an enginer I've trained for this moment my whole life