Best place for a switch when using an always connected power adapter

This is only an educated guess given the pictures and description. You need to do more testing and convince yourself that this works.

It looks like you could take wires 1 and 2 and connect them and get blue LEDs only, and 2 and 3 and get blue and white LEDs. if this is true you could buy a little timer relay kit from any electronics hobby store and connect it between wire 3 and what wire 3 plugs in to. Set the switch to "blue only" and leave it in that position, using the timer relay to do the white+blue cycle as you want.

People typically use JST-3 connectors to connect these strips because that is typically how they come from the factory. You can get more of these connectors from AdaFruit or Amazon or lots of other places.

If you are going to be soldering a lot of these connectors on, then I recommend getting a solder pot. You can then dip the tips of the wires into the pot, then the end of the strip into the pot, and then just hold the tip of each wire to its pad on the strip and touch with a soldering iron for a second. It goes quick.

I'd try to organize your cable runs so that maybe you hit a whole row of boxes horizontally with each logical string. This is really just about physically making the cable easy to deal with, so do it however works best.

For power, I personally would run a bus of of ~14 gauge black/red wire along with each physical string of strips. This cable should be able to support at least 1500 pixels if connected to a suitable power supply. About once every 100 pixels or so, I'd tie the power lines of the strips into the power bus using tap connectors. If you want to whole thing to be completely disassemblable, you could use Molex power connects between the boxes (or groups of boxes), just make sure the connectors can handle the full amperage you will be pushing though them.

I'd prefer to run each physical string into its own power supply rather than having one big one, again just to keep things physically manageable and to be able to use smaller supplies and keep your wires cooler. :) I've had very good luck with the Meanwell 5V DC supplies, but you can get very cheap supplies on amazon and alibaba.

For driving the data to the whole display, you could use something like an arduino but most of the code out there can only drive a single string so you'd have to daisy chain all of your strips into one long logical string. Yuo will also have to jump though hoops to drive that many pixels with an arduino's tiny amount of RAM unless your patterns are very simple. I'd probably use a beaglebone black running this fork of the popular ledScape code...

https://github.com/Yona-Appletree/LEDscape

This can drive up to 48 strings simultaneously and has lots of convenient ways get you pixels into it, including being able to use Processing for very fancy stuff.

If the data cables from the last string end up very close to the beagle bone, you can connect them directly to the bone's IO pins. Even though the bone is sending 3.3 volts and the neopixels want at least 3.8 volts, I've found that it almost always works fine. If the run between the bone and the farthest string is going to be more than a foot or two or you see glitching on the display, then you can make a very simple level converter from a couple of transistors and a resistor or you can buy nice (but expensive) ones here..

http://rgb-123.com/shop/

Note that they also sell the PixelPusher which should work for you, but is also a little expensive for my tastes.

Note that when you are connecting everything up, try really hard to always get the pixels powered up before sending a data signal into them otherwise you can blow the 1st pixel in the string.

Report back with your results!

UPDATE: Here are some photos of 1500 pixel panels I build using some of these techniques...

https://www.flickr.com/photos/bigjosh/sets/72157655896681772/with/19099344714/

Note that using a Beagle Bone also lets you use a Wifi to control the display from your cellphone!