Just to tell you where I am coming from on this - I wrote the book! (It's fun to say that.) "Residential Ventilation Handbook" It's been said pretty well already, but bath fans must vent all the way to the outside. Not just near an opening. Use insulated ducting to reduce the possibility of condensation in the ducting. Use 3 screws to attach the ducting to the hood. That will keep the ducting from turning or pulling off. Be careful that the screws don't interfere with backdraft dampers. (Cloth duck tape was made from the same duck material sailor's pants were made from. Not good at all for duct work.) If you can do it, transition from the 4" nozzle on the fan to 6" ducting. The flow will be much better. Point the nozzle of the fan in the direction of the exhaust hood. Air is lazy. It hates to make 180 degree turns. Here is a link that might be useful: Heyoka Solutions

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.

Thanks for the reply. Looking closely at the gable vent I would be foolish to put this up as the angle looks too exposed, possibly allowing water in, and there is no screen mesh from critters. The 4" just looks like trouble if installing on roof. Yeah I can order them from Lowes or HD but the roofers are here tomorrow to put the shingles on, so I need something in stock. Nobody has them. That's what happens when you don't plan ahead enough. Guess a normal T roof jack it is. Supposedly I can put a butterfly damper inside the sheet metal riser up to the jack later, from within the attic. Anyone know why it requests one? Don't all exhaust fans have their own damper anyway?

Happy to help. I've had a bit of this work done myself, in two different houses. In one, I had a Fantech BFRK 100 installed. There was an existing bathroom exhaust fan that worked poorly but I didn't want to put in a new grill. With the BFRK, you take out the guts of the old fan, keep the housing in place, and then connect ducting that runs first to the Fantech unit located a few feet away up in the attic and then from it to the roof passthrough. I had that done by an HVAC contractor who was on site replacing HVAC equipment. I'm not sure they would have done the work otherwise, it was like a favor to me. The second one was the replacement of a 20 year old exhaust fan with light in a WC closet that was noisy and ineffective. I had that done by an electrician who also was doing other work. He needed to go into the attic to remove the old unit, connect the new one to the framing, put in a new duct and connect the power. It made a huge difference. That one also vents through a roof pass-through, important because we use it to vent the bathroom of moisture from showers and baths. Good luck and keep your BS detector handy. If other exhaust vents terminate in the attic, it's time to fix those too. Each duct run needs its own roof passthrough unless you're using a Y setup with one fan at the base of the Y for two locations. Be careful if so, those can be tricky if the duct runs from the rooms to the base of the Y aren't of equal length. As far as duct run distances go, the unit will specify the maximum distance. For my most recent job, the ceiling unit with light that was 110 CFM, I want to say the spec for the duct run was something like 60 feet. I wrestled with putting a wall vent on a gable end but decided to stick with the existing through-roof vent.