It makes sense - look for "low voltage LED Lighting" to see what's available, or slide on over to Electronics to build your own.
Dimmable LED bulbs are those with drivers built-in that are "dimmable" - if just running controlled current to LEDs, they are all "Dimmable" (lower current, lower light) so, no, you don't want an LED described as "Dimmable" (nor one described as 12V, probably - a "bare" LED with no other parts attached is best when you are going to feed it controlled current.) If you purchase a "12V dimmer" pre-made, follow the dimmer-maker's suggestions on types of LEDs to attach to it.
LEDs are current devices, so controlling the current will give highly consistent light output (controlling the voltage, less so)
First I would go with warm white lights not the cool white, I think it makes a softer look to the bedroom.
In terms of the switching, yes a push button switch would work fine, you just have to make sure that the switch is within spec for the lights used. I have rather opted for a dimmable solution with my childrens beds. since they like to have some light at night. a good option here is this:
Its rated up to 8 amps, so if you look at the strip in your post, it consumes 2.5amps per 5m section used. (provided that you buy a transformer that can supply up to the required amps (ie 2.5amps per meter) -- you can buy a smaller amp transformer, but the lights will not be optimally bright at the highest level.
Your switches are only rated 3 amps, which means you can run about 5 meters of your spec LED Strip. -- Also to note, I would go with a stronger LED strip, generally 1 amp per meter is a nice powerful light (then you would only need a meter per child)
Another thing to consider, is the use of the 3color led's whereby you can have a color chooser.. but that may be getting too complicated.
Your series vs parallel question: You must definitely hook up the two lights in parallel, but just accomodate for the increased amps which will be required. The lights will draw the amps needed so you cannot really blow anything (its the volts which will blow the lights)
Finally: I find that the higher power led strips get a little warm, (meaning that the sticky tape sides start to get less effective over time). would be best to stick the strip to a metal surface if you could choose (it would act as a heat sync)
I actually ended up using a Chinese paper lantern to put the led strip in, it looked quite nice.
Best Answer
Mains voltage (110 or 220V) LEDs typically are sophisticated devices with internal power supplies that run low-voltage LEDs. The very point of the internal PSU is to provide stable operation which prevents controlling them with 220V dimmer.
As a rule of thumb, of 220V devices only incandescent bulbs are dimmable. All others (fluorescent tubes, leds) should be treated as not dimmable.
Your 12V PSU and dimmer are designed for 12V LED strips or G4 bulbs (which are also 12V).