Would it do this: run a bathroom fan based on humidity level?

The short answer is yes. The longer answer would depend on the type of hardware you have. Do you want the chime on your phone? Which phone do you have (Android, iPhone, Windows).

With this:
https://shop.smartthings.com/#!/products/smartsense-temp-humidity-sensor

And any of the zWave or Zigbee switches supported by smartThings.

And maybe this app:

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