Hi!

Lets say you have a pedal board and going direct to the mixer. The pedal board is all unbalanced and mono, but at the end you are using a TRS -> XLR cable to connect to FOH.

Is it correct to say that any noise that might be picked up on the way by the hot and cold conductor will be eliminated since the the cold conductor (without any signal, just noise) will first be inverted before the two signals are summed together? Just like with any balanced signal connected with a balanced XLR cable?

No.

No. Noise will not cancel out because you are sending an unbalanced signal down the line. Balanced signals send the information down both wires. Unbalanced down only one wire. Using a DI box is the best way to get an instrument-level signal into a mixer. It converts an instrument-level, unbalanced signal into a microphone-level, balanced signal.

Actually the answer is somewhat yes. Common mode signals will be rejected by the differential input. The problem is that the output impedance of each leg is probably different therefore the amplitude of induced noise on each line isn't quite the same. If you add a suitable resistor to the ground terminal of the unbalanced output, you'll get an "impedance balanced output" which will increase the CMRR of the line. A propper output transformer or balanced line driver would be even better.
Anyway, when connecting an unbalanced output to a balanced input, it's a good practice to use a balanced cable the way you have described. You can often even lift the ground if necesarry.

BTW while many people gon't realise, many large format analog consoles (Amek, SSL 4000 ...) had unbalanced or impedance balanced insert sends and balanced insert returns. Even some current microphones (e.g. Oktava MK-012) have an unbalanced output. The connection is still balanced, though.

cable length and associated loss of hf plus impedance mismatch and grounding issues are rather serious concerns imo...

Thank you!

That was what I was thinking!

Hmm.. if you lift the ground, wouldn't you lose the signal? There would be no complete circuit.

Anyhow, to the OP. Just use a direct box. They exist for this specific purpose.

I was testing some gear last week and one was a distortion pedal and was kinda surprised how much noise it had, nothing to do with being unbalanced..Simply a lot of hiss...

Not really. Check the attached schematic. If the input is transformer balanced, you'd even get a proper galvanic isolation. Even with an electronically balanced input, this would solve a ground loop as long as the potential difference is within the common mode capabilities of the input circuit.

Yes, I'd normally also use a DI box, because it would give more predictable results. But just for academic purposes, a DI box primarily does impedance conversion. Whether you need this or not depends on the drive capability of the output. If this was a piezzo or magnetic pickup, there would be no doubt that it's necessary. Considering that a pedal board is almost always made of active circutry, it's quite probable that it's capable of driving a longer cable and a typical microphone input, although pedal boards are usually used with higher Z inputs, so it could also have problems driving lower impedance load. OTOH it's also possible that the output of a pedalboard is actually better at driving low impedance than an active DI box as it isn't constrained by the very limited current supplied by the phantom power. As I've already mentioned, balanced cable driven by an unbalanced output and driving an balanced input behaves as a differential line. It is not impossible that it would perform better (noise, THD ...) without a DI. Keep in mind that with a passive DI, you are decreasing the output impedance by decreasing the voltage but the DI transformers usually have higher secondary resistance/output impedance than many solid state outputs (one of the best DI transformers available - Jensen JT-DB-EPC has Z out of 150 Ohm while a cheap circuit based on NE5532 op-amp can easily go an order of magnitude lower). This means that you are practically sacrificing level without gaining any drive capability. This also means that you'll need more gain on the preamp (more noise) and since more gain on the preamp usually means less negative feedback in the preamp circuit, distortion and frequency response of the preamp would also get worse. Whether a transformer balanced output of a DI would increase interferrence rejection more than the decrease in the level would make the signal prone to interferrence is also not certain.
BTW MADI is an 0.5 V unbalanced 100 MHz signal and still easily runs 100 m, 3G SDI is a similar 3 GHz signal and still easily runs for a few dozens of meters. All you need to do is make sure that there is no too much of a potential difference between the devices.

TL;DR
A DI will almost certailny sound OK, but depending on the output circutry of the pedalboard, it might significantly increase the quality of the signal but could also slightly decrease the sound quality in some cases.