Wiring LED strips in parallel & ground
I think you are overestimating the size of wire required to attach to the LED strips, I have connected a lot of single color LED strip (in excess of 1mtr) with the wire that was already attach and it was reasonably light gauge.
Look at this Example on Ebay and you can see what i mean.

I am note an Electrical Engineer so i don't know the method to calculate the minimum wire gauge, but if in doubt contact the manufacturers and ask them or search on the web.

As for the Transistor mounting/wiring you would mount that on a small circuit board with terminal block fitted for ease. Any power/connecting wire should not be less that used to the LED strip.

Also you will need a current limiting resistor attached to the Base pin on your transistors. You will have to work out the value of that based on the spec sheet.

There is a bit of learning required first, once you have researched the above you will have you exact answers.