1/4" TRS cross-coupled output with bypass capacitor. Is TRS jack wired for cross-coupled or not?
The main outs are cross-couple, capacitor bypassed, single-ended output, for a TRS jacks.


Are you sure you got an engineer and not a sales person?  My best guess is that it's floating and AC coupled.  But you can't have a floating, single-ended output.  See here.


Floating means that the average between Hot and Cold can wander wherever it wants to go as defined by the external connections while the difference is tightly controlled to represent the signal.  By tying Cold to Ground, you then cause Hot to act like a single-ended output, but then it's no longer floating.  Or, if you leave Cold unconnected and connect Hot to anything, then Cold will take most of the signal (inverted) leaving very little for Hot.
AC coupled for your purpose means that it will naturally adapt to whatever DC offset there might be in its own output driver or in the receiving equipment.  This is important because speakers and transformers really don't like DC.  This doesn't affect whether it's floating or single-ended or whatever else because the capacitors that do this are big enough to act like plain wires for audio.
Single-ended means that only one wire carries the signal.  Hopefully you can see now why "floating" and "single-ended" are incompatible.
Bonus, useful soon:


Balanced means that the two signal wires have equal impedance so that they pick up noise equally.  The noise can then be subtracted out at the receiving end to give a clean signal.  Equal and opposite drivers are not required; in fact, the board in the youth center that I volunteer sound for has a bunch of XLR and TRS outputs that connect Cold to Ground through the same sized resistor that connects Hot to its driver.  As far as noise performance goes, that's still balanced.
dBFS is a decibel scale that is referenced to the maximum possible level (FS = Full Scale) and is therefore always negative.  0dBFS by definition is just barely not clipping.



Now that we have some definitions, I can now tell you that, unless you have a very very low-end board like the one in my small "fun" system at home, then at least the main outs will be balanced, unlike your RCA input.  So you may have some noise issues, gradually becoming a certainty as you increase the length of your cable run.

If your amps are far enough from the board that one operator can't reach both at the same time, then I would strongly recommend using a balanced input for the amps, even if you have to buy different ones.  If they're close, or if you insist on using the unbalanced input anyway, then there are still some solutions:


If your output is floating, then your solution #12 should work, connecting Hot to Signal and Cold to Ground, as long as you never exceed -6dBFS.  The reason to apply this new ceiling is because you actually lost that much headroom by making those connections, which is not reflected on the meter(s).
If your output is not floating, then I would recommend your #10 instead, connecting Hot to Signal, and Cold to nothing, which allows you to go all the way up to 0dBFS.
If you can't determine whether you're in my #1 or #2 and you're okay with spending some money, you can use an audio transformer between Hot and Cold on the primary winding and between Signal and Ground on the secondary, which also allows 0dBFS and gives you a balanced receiver if you put it at that end of the cable.  Be aware though, that good audio transformers are expensive!  For example: http://www.jensen-transformers.com/transformers/line-input/




One more thing I'll mention even though you didn't but because I've seen it done before by people who don't fully understand: DO NOT run speakers through wire that's designed for microphones!  This may seem obvious, but I've personally seen several cases where a 16-channel snake, for example, was used for 12 channels of mics and 4 speakers.  The amps were next to the board and the speakers were up front, using the 30-ish-gauge wires to push potentially 200 watts through 8 ohms, which comes out to 5 amps...through 30-ish-gauge wires.  Not good!