Arduino Mega 2560 not recognized/overloaded
First off, that is one heck of a rig.  I'm impressed to the point of "I gotta get me one of those."

Bundling your wires shouldn't have the kind of impact you are seeing, unless there are bare wires touching.  You might get induction interference, but not a total shut down.

From what I can see here, it looks like there's so much stuff connected to the Arduino that it's probably starving it for power.  You should look into using power rails for Gnd and +5v.

You will also need to check if you are supplying it with enough amps, too.  Your computer's USB port only supplies 0.5A, unless it happens to be USB3, then it's 0.9A.  Even your 5V1A PSU might not be enough.  Consider finding a 2-5 amp wall wart to do some more testing with.  There are usually spec sheets with components, so you'll need to add all the power draw for all the components, including the Arduino, and make sure you have more than enough to meet that calculated power draw.  Exceeding it by at least 10-20% is usually considered good practice, even for a test rig.

Consider powering your rig's new power rails with the extra PSU even when the Arduino is plugged into your computer.  As long as you make sure you aren't trying to take power from the computer and the PSU at the same time with the same component, you should be fine.  (Don't cross the streams.)

There could be a short or feedback loop somewhere in your sensors that's causing the extra drain.  There are so many wires here that it'd be easy to hook up something backwards.  Along with the number of wires, the bundling doesn't let SE-us help you track that down.

You might want to look at the direction your pins are facing.  It was an odd situation for me, but having two of the metal clips facing each other was enough for a short.  In your "Relay Portion:" pic, they are all facing the same way, which is good.  See if there are places where that's not the case and fix it.  Like I said, it was an odd situation, but it's a phantom problem you can avoid.