The two resistors are used in series with the pin, but should be used as pull up resistors (NPN sensor).
So connect sensor outputs directly to the Arduino pins, and connect the resistor between pin and 5volt.
Educated guess is that the relays are using 80mA each when active.
That, and the draw of the Nano and sensors (~400mA) is just withing the limit of the USB supply.
Maybe wise to connect the 5volt supply directly to the 5volt pin instead of running it via that tiny USB diode
(that diode also lowers the supply by ~0.4volt).
Interference:
Don't run the fan wires next to the low voltage wires.
You might have to use snubber circuits across the relay contacts.
A VDR/Varistor, rated for your mains voltage, could be enough.
The relays itself might be rated for your mains voltage, but the board might not be.
YOU are the judge of that.
You might have to ground any exposed ground/controls/etc. of the Nano to mains ground.
Ask a electrician if you're not sure.
Leo..