3A lamp on 5A outlet
You should be OK to use a 3A appliance on a 5A circuit. The circuit is capable of supplying up to 5A but it is the individual appliance (lamp in this case) that determines how much current is drawn.
Round-pin plugs vs rectangular-pin plugs
Modern usage
I'm surprised a new home would have BS 546 round-pin plugs instead of the more usual BS 1363 rectangular-pin plugs.
Wikipedia says
Although BS 546 plugs and sockets are still permitted in the U.K. and are used for special purposes, new installations since 1947 generally have used BS 1363 sockets and fused plugs.
Until now, I thought the only place where round-pin plugs were used was in theatrical stages, where I think they are used to prevent people plugging inappropriate devices into theatrical lighting circuits.
Safety
If the BS 546 plugs don't have built-in fuses, they don't provide any protection to the line-cord between the plug and the lamp.
BS 1363 plugs have additional safety features which may not be present in BS 546 plugs
- fuse to prevent overload of appliance cord (& hence fire.)
- shutter in socket opened by earth pin (safer sockets).
- insulating sleeves around base of live and neutral pins.
- high insertion force to reduce risk of partial removal.
- designed so earth wire is last to break under cord strain.
For example a small child can poke their hands (or part of a metal toy) behind a loose BS 546 plug and be in contact with 230 V.
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
Honestly. Your best approach is to consult with IKEA about powering the lights from one single common low-voltage transformer/supply.
Nobody, not even Ikea, thinks a bunch of chunky transformers is the way to go. They did it that way because that is the lowest-cost way to sell the lamps in a single unit and still get the items listed by your regulatory agencies. They may even have an Ikea-approved way to use a common transformer.
Generally, low voltage power is much more lightly regulated, so the low-voltage cable run from wall-wart to lamp proper is thus less regulated. That's still a good idea, but with one common transformer if possible. If that transformer has a cord, feel free to put a 5A receptacle and <=5A fuse.