The only choice I see is to use an unbalanced cable to the unbalanced input on the amp. There is no advantage to try to adapt your cable to the balanced input.
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My amps ONLY have balanced connections as they are fully balanced and do not provide for single-ended conversion. Prior to my current balanced preamp, I used several single-ended preamps. I also used Jensen input transformers to effect the conversion. Although my transformers were an outboard model requiring an additional very short IC cable, Jensen or Lundahl transformers are used by some manufacturers "inside" of amplifiers for the same reason. The transformers offer the added benefits of a ground break (they can correct hum problems) and I believe they also block DC. However, there are impedance matching issues requiring you to keep the IC cables short'ish and they may not work with amps having low input impedance....like 10-20K ohms.
I was in the process of upgrading and testing equipment.
Allnic L5000 DHT to Allnic L10,000 OTL/OCL (no capacitor, no transformer)
The signal fed a Allnic A6000 300b monoblock with both balanced and unbalanced inputs. However, the amps are single ended, unbalanced circuit design. I had run balanced because of the 35' run to the amps. When I switched to the new preamp, the balanced output of the L10,000 required the use of an internal quality transformer. Notably, that transformer removed the magic the linestage preamp offered. So I sold my cable off for unbalanced solution and it was the single best move ever.
In conclusion, I reach out to the manufacturer of each component regarding circuit design and if the upstream component is true balanced circuit design, then I used balanced cable and ideally components downstream that also have true balanced circuit design.
In this case, I did the opposite. Everything from DAC to amps were unbalanced.
Getting back to the OP's question as I read it....
Unbalanced out to balanced in ( with an adapter) is quite simply unbalanced. Balanced lines have two cables, operating 180 degrees out of phase (push-pull) . The balanced input essentially measures the difference (sums them up). The big advantage is that any noise that is common - like noise picked up from RF interference or 60 Hz interference, ought to cancel.
If you run only one of them, its exactly like the unbalanced.
One caveat - this assumes that the amp’s in is true, inherently balanced meaning there is a differential input and you use either all or half of it. In this case you would incur no additional conversion circuitry. Sadly, more than half the unit I have seen actually convert balanced to unbalanced and then rn through the normal circuitry. In this case, unless you have a noise problem, balanced is inferior unless you actually like the distortion another stage gives (admittedly small if well done, but in theory, its there).
So bottom line - it should mostly sound like unbalanced since it IS unbalanced with minimal or no differences.
To your 2nd question, no specification will tell you.
And manufacturers don;t want to say " i have a single ended circuit and plopped an op-amp (or 3, to make a classic Instrument Amp Configuration) on it to convert it to balanced". Or whatever. But often it is true.
And again, aside from noise immunity (and 3dB more gain, but that's also mostly a noise benefit), there is NO DIFFERENCE between unbalanced and balanced.
On most of my designs the input is a differential amp anyway. So they are the best candidates for balanced. And even with my stuff, the only difference i know of, or hear, is noise immunity; unless one runs the amps as fully balanced monoblocks, and that's an entirely different situation where you get more voltage (power) too.
Moral of the story: don't sweat it. If you have noise pickup, route the wires away from the noise.