Have a look at my drawing:https://imgur.com/EYQ2HAY
3) What kind of cable to I need to bridge the data connection between the two parts?
Do I just solder these cables directly onto the strip?
No that site wants to infect my computer with all sorts of tracking software so we like you to post images like you have direct to the site.
Does that look about right now?
I have included the transistors resistors and capacitors in there as well.
Not sure how thick 10 gage is, is that AWG or SWG?
I would like to run just one power cable parallel to each strip so that I can tap in here and there for more power. Do you think this is good?
Or can I get away with a thinner cable?
I am sure you can. I would say you could get away with 14 AWG.
Your diagram is almost correct. It does not emphasise however that the data line is always paired with the ground. So the power run - 5 V and ground - to the Arduino should not be a separate run; it should run from the start of the LED strip back to the Arduino as one bundle with the data line. This 5 V line is not carrying any substantial current so it does not need to be such heavy cable however if you run it back from only one LED strip (as you only need it from one) then the ground travelling with the data from the Arduino to both LED strips should be reasonably heavy as you do not want voltage drops due to the current drawn by the LED strips.
Not sure, though, what I should enter for load current.
There's a lot to process in that paragraph. Do you mean something like this?
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?
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.
I read in a couple of tutorials that you should supply power every 6.5' - 9.8'.
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.