It might be too small for an HVAC person to take it on. It's somewhat hard to recommend because it does involve wiring, and perhaps some siding and flashing work. Many locales require electrical work to be done by a licensed electrician, and you don't want some bozo not doing the flashing correctly if they mess with your siding. You should be able to run the exhaust duct through the joist bays that make up the bathroom ceiling and through the rim of the cantilever. It could exhaust downwards out of the cantilever soffit, but that can sometimes not be the best choice. A small remodeling outfit might be well-versed all aspects of what you need to have done. A capable and conscientious handyman? That can be good or bad, depending on the person. Consider asking for, or use these as points within the conversation: -Rigid duct. Nothing "flex" or with ridges/corrugations. They can use a small section of flex to connect the fan housing to the run of rigid, but there should be no tight radii in the flex or any kinks in the flex. -All seams in the duct sealed with aluminum foil tape. Not "duct tape". -Insulation around the duct, at least within 4' of the rim. -Any horizontal runs in the duct will be pitched down for drainage in case the moist air produces condensate within the duct. -Ask how they will flash the damper into the siding of they run it through the rim. -Ask how they will damp the exhaust if they run it downwards through the cantilever soffit. -Consider getting a timer switch for the fan. You turn the shower light ON, the shower light and exhaust fan turn on. You finish showering and turn the shower light OFF, the light turns off immediately, but the fan continues to run for a time set by you; another 20 or 30 minutes. Do get an idea of how many feet of duct you'll need. Fans are rated to move a certain amount of air, the longer the run of duct and the more turns within that run, the more resistance to flow and the less air the fan can actually move. You might need to up-size the fan to actually move the needed amount of air.

I would tend to agree with cobraguy about the fan running in reverse. If it's drawing in air and the vent cap is on the roof it's most likely getting the smell from the vent stack. I have a roof level deck on my house and can sometimes get the smell from one of my vent stacks. If you can access the fan in the attic, disconnect the hose and with the fan running check the air flow. It should be blowing out of the fan housing and into the attic. Most vent caps have flappers on them to prevent back flow so that might not be the issue.

there are two separate issues that need to each be addressed entirely on their own. in all of this, remember that your fan duct is a chimney and reacts accordingly. you usually have warm air flow and waste even when not "using" it. if you have condensation and it's all in cold weather, insulate the thing. Not with that trendy gimmicky thin plastic junk but with a whole lot of plain old fiberglass. a lot of it. reseal any seams first, perhaps with brush on goop and fiberglass mesh. the water damage is a separate deal and I would not be any way qualified to suggest anything concrete without eyes-on. both issues do seem to be from poor installs. the s-trap is funny, though.

They sell smoke kits to test for things of this nature, problem that you're not thinking about is that smoke is naturally light and is easy moved from one location to another with the use of a fan. Heavy air, such as steam is not. Seeing steam vented could merely be only a portion of the steam vented from a room and due to exterior conditions being very cold in relation to the room that is vented. Just seeing steam vented isn't an indication of how much in proportion of what is in a room.