There's a lot to process in that paragraph. Do you mean something like this?
Well, not quite.
You show separate wires connecting the ground of the power supply to the Arduino and the data to the strips. You need to
always run the data and ground wire together. Since the data wire goes from the Arduino to the strip, the ground must go with it
as a pair from the Arduino to the strip. Since you have a data wire to each of two strips, then
a ground must go with each data wire
as a pair from the Arduino to
each strip.
Now you want 5 V power for the Arduino. This means that you need to run the 5 V line
as a triplet from the start of
one strip back to the Arduino along with the ground - and necessarily, the data wire also. Ground and data always travel together, and Ground and 5 V always travel together. Three is not a crowd.
Also, I read in a couple of tutorials that you should supply power every 6.5' - 9.8'. So I should be pretty solid with supplying power at the beginning and end of my 13'2 run as well as the beginning of my 7'3 run, correct?
No. If you read in tutorials that you should supply power every 6.5' - 9.8, then 7'3 is
in that specified range and so "
every" 6.5' - 9'8 means at two points 7'3 apart, which means the beginning and end of that strip.
And since 13'2 is twice that length, then "every" 6.5' - 9'8 clearly means the first end, 7'3 into that strip and the other end.
It will be challenging to make multiple connections since all of this will happen in a very tight, recessed, channel. So unless I absolutely MUST make more power drops I would like to avoid it.
It should not be difficult. It means baring the 5 V and ground at that point for a few mm, soldering a jumper the few mm to each of the two pads on the adjacent strip, and ensuring there is insulation between the two wires. A wedge of plastic or PCB substrate material, and covered in epoxy.
Why all this talk about keeping the wires together? It's called a
transmission line. We are talking about
radio frequencies here, similar to those used on your AM radio. Wires that are not kept together are called
antennae, used to transmit
and receive - in this case, RF interference and impulse surges.