So the LEDs are separate from the controller, right? Do you want to mount the LEDs in little boxes, or in the stroller?
For mounting LEDs, I've used hundreds of these
over the years.
2. How should I handle to cabling, (is it better to solder everything or use sockets)
It's your project, so that's up to you. For stuff that's external to your main contoller box, I'd use connectors. Everything inside the box can be soldered.
Look at how a computer or a piece of stereo equipment is made... Everything plugs-into the main unit, except sometimes the AC power cord is permanently attached. A mouse or keyboard has a cable permanently attached, but usually a printer does not. (If you look inside your computer or stereo, you will see lots of connectors, but for a simple home-built project this usually isn't necessary unless you have multiple circuit boards.)
If you are not using a pre-built Arduino board, I'd put the Arduino chip in an IC socket. In fact if there are any other ICs, sockets are nice if you have a problem... You don't always know if you've got a bad/blown chip or something else, and having a socket makes troubleshooting/repair a lot easier!
3. I don't want to use my Mega to run it, so I bought two arduino kits (quarts, chip, ceramic resistor, etc)
How's that going? Did that kit come with an Uno board, or just an Arduino chip?