Outdoor Power - Do not power the camera using an outdoor outlet. You have all the problems you listed and you make the system more susceptible to tampering (an intruder can simply unplug the camera and blind the system).
Powering the camera - Let's begin by checking if the camera and its power supply is rated for outdoor use. The exposure to moisture, and to a lesser degree extreme temperature, makes many indoor units unsuitable. The connection of the power supply to the camera needs to be water resistant. If it is not, you need to consider a different camera unit.
In general, outdoor power lines need special cable or a metal or plastic pipe to protect the cable. The power to the unit is apparently low voltage, so it may not need the same level of protection as it would if it were 120 volts.
You will need to bring the wire through the wall (or through a soffit) at some point. You want to drill through an area that DOES NOT contain framing. You will go through siding, sheathing, probably insulation, and then finish material, probably drywall.
The main issue is sealing the entry holes from water and air infiltration. This can be done on the outside using a good outdoor caulk.
On the inside, you can use a low voltage junction box to protect the wire and then route it to an outlet. The best approach might be to run the wire to a double box, low voltage on one side and line voltage on the other.
You could then have the wire exit the front of the low voltage side and have the transformer plug into the line voltage side.
Mounting Camera - Most modern cameras (at least those without heavy housings or motor mounts) are fairly light, just a few pounds, if that. They can be screwed into the sheaving of the wall, which is located beneath the vinyl siding. You need to drill through the siding and into the sheathing and then screw the base to the sheathing. As you are mounting, you can seal the screw holes and even the base of the camera with outdoor caulk.
If the camera is heavy, you should screw the camera to a framing member (stud). These are found at corners, regularly spaced horizontally (usually every 16 inches from a corner), and near the roof and floor line of each level. You can use a stud finder to locate them.
That is called fan-fold insulation. Typical for under vinyl siding in older homes.
Very surprising that there is no sheathing, but not unheard of. What I would do is get a round cut-in style fixture box and mount it directly to the vinyl block. This seems like your only option, apart from removing some siding and replacing the vinyl block with a fixture box style piece from Arlington Industries.
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http://www.aifittings.com/catalog/siding-mounting-blocks/siding-mounting-block/
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As far as the NEC code is concerned You can run the UF in the void between the furring strips it sounds like you will be fishing the wire into the void. If you have to remove the siding to put it in staple your wire 1-1/4 inch away Or more from the furring strip. Exhibit 300.2 has a good example of this. Some areas do require conduit Chicago just about everything requires conduit or metal clad. But by the standard code UF would be allowed unless additional requirements mandate conduit.