I want to mount an IP camera in a room. The power outlet and Ethernet ports are near the floor and I want to mount the camera near the ceiling. I was thinking of cutting through the wall at the top and bottom and adding a recessed low voltage cable plate. I read that it is against code to run high voltage power wires through the wall, but the power wire I'm planning to run is an AC adapter on one end, lowering the actual wire voltage to a handful of volts. Is that allowed?
There's a better solution to this problem
Most low-voltage wall adapter outputs for low-powered ITE such as an IP camera qualify as Class II power circuits under the NEC, which means that Article 725 applies to the situation you propose. However, the wire used for wall adapter output cables is not legal to stuff into the wall in the way you are proposing, as it's not a listed UL/NEC type CL2 or CL3 cable as required by NEC 725.135(A) and (M) (instead, it's some AWM type, at least on the adapter I had laying around handy).
So, you have two options at this point:
Run a CL2 cable through the wall
The perhaps-simplest way to do this would be to run a listed CL2 or CM cable through the wall -- two-conductor thermostat or in-wall speaker wire works, although you'll need to get jacketed cable instead of the normal unjacketed-pair stuff in order to comply fully with 725.139(E). At each end point, you'd terminate it to a dual binding post or dual banana jack wallplate (you can get modular wallplates that can be set up this way -- look at the Leviton QuickPort line for one of many example). You'd then chop the AC adapter's output cable in two, fit banana plugs to the chopped/stripped ends, and plug them in in the appropriate places, minding polarity of course.
Just make sure that the low voltage plate is on a separate bracket or box from the 120VAC, or that a listed divider separates the 120VAC gang from the low voltage gang.
Do it the old fashioned way
The "old fashioned way" is to use a proper Chapter 3 wiring method such as NM (Romex) cable (note, NOT flexible cord, or else it's as many lashes from an extension cord as it takes to acknowledge that NEC 400.8 is true, wise, and holy!) to run 120V power from a nearby circuit to a new receptacle located near the camera's home, which then can have the AC adapter run to it.
You'll of course want to turn off and figure out the circuit you plan to use for this, and use proper splicing and cable securement techniques. Also, the power cable can't share a junction box with the Ethernet cable unless you shove a listed divider between two separate gangs, one for the Ethernet side and the other for the 120VAC side.
It's always safest, to consult your local building commission, when unsure about code restrictions.
At the very least, even if it isn't against code, which it probably isn't - you should exercise no less care, in ensuring that it properly routed, and protected from damage.
Outdoor cameras that come with power/video cables, often have thicker jacketing, for such installation concerns. While wall warts on the other hand, usually do not.
Also, POE doesn't necessarily have to be built-in either. There are splitter/injector kits available, that convert two cable setups, to one.