Pretty hard to miss wood smoke fumes if they are coming into the shed. It's not like most other combustion exhausts which are more or less odorless and colorless. And a smoke detector does a good job of letting you know if there is a leak if you don't notice, so a monoxide sensor may be superfluous. Plus a smoke detector can also warn you of a fire, which is a more frequent hazard with a wood stove.
And yes, an external wood stove hot air system will definitely dry your house or shop out. It raises temperature at the lower relative humidity level of ambient air, thus drying it out substantially. In winter at freezing temperatures, the air is normally low in humidity anyway. So it dries it even further.
On the bright side, It's a plus for preventing rust on machines. But not so good health wise.
Most people with wood stoves put a kettle of water on top of it to help humidify the air. I'm sitting in front of one now as I write this. Adding humidity to vented hot air would be harder, but not impossible. In deep winter where the indoor humidity can reach as low as 25% here, we not only use a kettle, but also a humdifier running 24/7. That often only brings it to 40%, which is still low, but better.