Well, in theory, assuming wiring is done right the ground prong will contact the ground terminal in the outlet and that terminal is connected to the ground with a thick enough wire and touching it is as safe as touching a washer or fridge chassis.
However you should not do that ever, because Murphy is out there...
First of all, the ground prong is very close to the phase and neutral prongs and there's a slim yet quite real chance that you will touch the phase prong at the same time. Once this happens you get a ground fault and if there's no ground fault protection that yields a major short circuit which gets you a nice several millimeters deep burn on your finger (I have nice experience touching phase and neutral terminals together with the same finger - heals in two weeks and can be fatal in an unlucky electrical setup). If you're really unlucky (like you're touching a grounded appliance at the same time) you get a ground fault through your body and this yields fatal shock. Such things do happen - it's just a matter of time. You do that no-problem one thousand times, then you accidentally do it wrong and consequences are dire. Chances are slim, but quite real and potential consequences are serious.
Second, you can't be sure at all times each outlet is wired right. Wiring may fail at some point and put you at risk. You can also try to do that in other person's house where wiring happens to be done wrong and that puts you at risk.
Finally, there's this cheater plug thing. Clearly if you plug a three-pronged device into a cheater plug and then try to plug the cheater plug and habitually try to find the prong you likely find either phase or neutral prong and that puts you at risk.
So the bottom line is you technically can do that, but you really should not - getting such habit is potentially very dangerous.
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
Based on the Lusa Lighting Model #33000 which are sold under Hampton Bay brand #148652 (For the 3 Bulb White kit), these are not LED lights. They are straight Halogen Light Fixtures.
Each of the Light is a standard 120V Halogen Bulb in a fixture, wired with a standard 2-prong outlet plug. The kit comes with a basic 3 to 1 power strip.
If you have 4 lights, 80W / 120V = 0.667 Amps, you can just use your own power strip instead of the included one. If you have 4 kits of 3, you need up to 12 outlets. Even with 12 20W fixtures, that is still only 240W, or 240W / 120V = 2 Amps.
You could wire them in parallel, but a Power Strip already does this for you, and is safer than wiring it yourself if you are not experienced.
Otherwise see How to connect multiple light fixtures to one switch?