First, everything woodchips told you in his answer is absolutely correct, and you should follow his advise about solving the cause of the excessive moisture. Your fan should run for 5 to 10 minutes after using the shower to clear the moisture.
With that said, let's examine the paint problem. I think there are a few factors causing the paint finish to fail. First reason is that moisture is obviously penetrating the paint layer and causing the paint to separate from the wall board, thus peeling or bubbling occurs. This is usually caused by a lack of a proper sealing/primer coat on the wall board. Today, we would use a moisture resistant drywall or mildew/mold proof paperless drywall, but even that would have to be properly sealed to prevent a finish paint failure. The primer has two functions. The primer seals the surface of the drywall to prevent moisture from being absorbed, and secondly creates a strong bonding layer for the paint to adhere to.
In your case, I suspect a couple of things happened. First, the wallboard was not primed properly and most likely a very cheap grade of paint was used directly on the wallboard and mud joints. (thus the rust spots showing through from the fasteners)
Follow woodchips advise and remove the problem areas. If some patching of damaged drywall surface is required, use a setting type drywall compound rather than general purpose compound. Setting type is less susceptible to moisture decay than regular compound. Vinyl spackle can also be used if the spots are small. The next step would be to properly prime the entire ceiling. I would highly recommend using an alcohol based pigmented shellac such as Bin's Bullseye primer. This product will seal the surface and block any moisture stains from bleeding though again. If the staining is very dark or wide spread, two coats of Bullseye would be a good idea. Gently buff sand and remove any dust from the freshly primed surface before painting. When painting, use a good grade of mildew resistant paint. I would use a satin or flat finish enamel instead of common flat ceiling paint. An enamel will give you a much more durable and moisture resistant finish. It will also bond very strongly to the primer.
Side note: Be sure the fan is vented to the out of doors. I have seen them vented directly into the attic space. Excessive moisture above a ceiling, especially around a cold soffit can cause moisture to migrate down through the ceilings close to an outside wall. This usually causes finish failures along the ceilings on an outside wall. Likewise, if a soffit area is very cold, moisture will condense on the interior ceiling surface along the outside wall from the inside and cause staining and finish failure.
In this case, it looks like you'll have to run a new 20A circuit. As per Article 110.3(B) of the National Electrical Code (NEC), and the manufacturers instructions.
NEC 2008
110.3 Examination, Identification, Installation, and Use of Equipment.
(B) Installation and Use. Listed or labeled equipment shall be installed and used in accordance with any instructions included in the
listing or labeling.
On page 5 of the manufacturers installation manual (PDF), it clearly states that you must use a 20A circuit.
Best Answer
If you have a vapor barrier (e.g. sheet polyethylene) behind the drywall, then you have a major problem. Painted drywall is airtight but not vapor-tight; moisture can migrate through it. And when moisture migrates through the drywall and hits the polyethylene sheeting, it has nowhere else to go. The fact that it currently condenses on the walls tells me that the walls are cold. That means the polyethylene behind it is cold, too. That means water vapor that gets through the drywall will inevitably condense on it and drop to the bottom of the wall, eventually rotting the whole thing out from the inside. Not good. Mold will probably not develop on the drywall at the top of the wall because that's not where the water will be collecting; it's the bottom of the wall you should inspect.
Vapor barriers in walls are generally a bad idea, but if there is any place you could safely have one in a bathroom like this, it would be on the inside, in the form of a water- and vapor-right waterproofing membrane, like RedGard.
If you are unwilling to rip out this assembly and redo it properly (with Redgard-coated cementboard instead of drywall, and no plastic vapor barrier inside the wall), then I would highly recommend preventing moisture build-up in the bathroom in order to preserve the integrity of the structure.