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You are technically correct. We goofed. UL-listed bathroom exhaust fans (with or without lights) are exempt from the rule requiring ceiling fixtures to be a minimum of three feet away from a shower or bath enclosure.
However, because my audience extends beyond the awesome power of the United States NEC (National Electrical Code), I must qualify that I am unsure this statement is universally true. To be more "helpfully vague", I should have instead written that "some electrical codes MAY prohibit electric-powered fans in this location". Sorry for the misunderstanding… just being overly cautious.
By the way, I am unaware of any bath fans that have a GFCI in them to prevent electrical shocks. The approved fans are instead designed to have no exposed electrically-conductive surfaces. If necessary, a GFCI would have to be installed separate from the fan, typically in the form of a GFCI circuit breaker or in the wall switch/outlet supplying the fan its power.
Thanks for your astute observation!
Most of the info you need to install the fan is in the included instructions, regardless of brand. The manufacturer should give you information concerning issues such as the size of the hole needed in the ceiling, and whether or not you should put insulation over the fan.
Locating the fan is dependent on the position of the ceiling joists. Having access to the attic makes it easier, of course. Pick an approximate location, and push a nail through the ceiling. You can find the nail above, or be really efficient and push a coat hanger through the hole. This will make locating the hole easier, since pushing the coat hanger will move the insulation (if you have a friend to help).
Since you live in an area whose climate requires lots of insulation, any restrictions in covering the fan can be overcome by building a small wooden box around the fan to keep the insulation from touching it.
As far as venting, you can vent either way. However, I would lean towards venting through an outside wall rather than the roof. Roof vents are always a potential leak waiting to happen, plus a heavy snowfall might, under the "wrong" conditions, block the vent flap making the fan nonfunctional.
Since there is a tendency for moisture to collect in the vent hose, it is important to cover the hose with insulation to keep it as warm as possible. If you bury it under the attic insulation, that should be sufficient.
I would have suggested a heat exchanger as was your first thought, but I agree with you… they can be expensive. Since a heat exchanger is more for energy conservation than air quality and thus removing the energy aspect from the equation, simply opening a window to keep some fresh air circulating is a cheaper alternative.
Since mold and mildew need moisture to grow, decreasing the airborne moisture in your mobile home is a more direct way to lessen the problem. A properly sized dehumidifier is one effective way to do this.
If you use gas for cooking, you may not be aware that one of the major byproducts of the combustion of natural gas is water vapor. Installing a vented hood over the stove can help remove some of this vapor as well as the moisture from boiling water and general cooking. Also, installing a ventilation fan in the bathroom will remove the moisture there.
Paintable surfaces can be growth mediums for mildew, so it is imperative that you use only mildew resistant paints in any high moisture area. Benjamin Moore and Zinsser both manufacture special paints for this purpose.
If you are sure there is no external water source, then you problem is definitely condensation. Whenever moist air moves through a duct, the moisture will condense IF the surface of the duct is significantly colder than the air. When dryer vent hoses or bathroom fan hoses vent into the attic, the amount of moisture that condenses can be great enough to cause water damage should it leak back into the house.
The best approach is to minimize the condensation by insulating the duct as best you can. If it runs across the attic floor, cover it with Fiberglas insulation. The duct does not have to be tightly wrapped... just protected from the attic cold. Ducting that goes through the roof should also be wrapped, though if the distance traveled through the cold attic is only a few feet it may be unnecessary.
Don't forget to put insulation over the body of the exhaust fan, too. Be aware that some fans require a little air space around them, so don't pack insulation tightly around it... just gently lay some over the top!
Another fact is the longer the duct, the greater the condensation. Examine whether you can shorten the length of ducting by relocating the vent. I know this may not be possible, but it is worth a look anyway. You might install a standard through-the-wall dryer vent kit or even a through-the-roof kit. In all honesty, I am not big on venting through attic windows or soffits. Not only do the window-soffit screens restrict air flow but they are likely to get clogged with dust very quickly. Besides, much of the moisture will be blown back inside the attic since the flow through these vents is often inward, not outward!
Replace flexible ducting with solid ducting wherever possible. A solid duct's smooth walls produce less restriction to air flow than a flexible duct. The damp air moves more quickly through the ducting, cooling less and thus taking more moisture with it. Your fans will also be more effective and efficient.
If the condensation is excessive, you must build drainage into the ducting. The first step is to slope the ducting so that any runoff exits your home via the exterior vent. This will require a little planning because you don't want the runoff to damage or stain your home. Select and install a vent that will keep the drips from running down your siding. Also, make sure the sections of duct are overlapped correctly with the male end above-slope of the female end... this prevents dribbling and makes sealing the joints unnecessary.
Finally, you should plan on reexamining the ducting in the spring to be sure there has been no accumulation of water over the winter.