Venting a new bathroom through the side of the house
If you are referring to a vent for an exhaust fan in the bathroom, then the answer is yes. I recently purchased a new home and the bathroom needed a full remodel due to mold and water damage. There was not an exhaust fan and we wanted one in the bathroom to help pull moisture out and keep the bathroom well ventilated. Previously I had helped a friend install an exhaust fan and we routed the vent out to the soffit. In our new house the soffits are a long way from our bathroom, so the easier route was to install a wall vent outlet, similar to a dryer vent outlet, directly below the attic vent on the side of the house. Make sure the vent outlet has flap inside to keep air from flowing the wrong way when the fan is not on, especially if the vent is installed on the windward side of the house. We painted the cover the same color as the exterior and unless you are really looking for it you can't see it easily.
Yes. Most lower-level bathrooms in my part of the world vent through the rim joist, and I personally have installed one at head height where there were no other options. There are no problems inherent with this approach, provided common sense is heeded related to where moist/smelly air is directed.