You've tried all of them (and a few that aren't), so you might be wondering what they're really for anyway:

1. The 13/14, 15/16, and 2TR_IN (tape input) connectors are inputs of course; they're not going to help you. The USB playback in that version normally connects to 2TR_IN, through an inaccessible mixer with the RCA's that then feeds the volume knob that is the only control between the input and the master fader. (For that reason, I don't like tape inputs or most effects returns. Give me some more channel strips instead, even though they're more expensive.)
2. REC_OUT (tape out) is apparently meant for recording. It's the main mix, pre-fade. Basically, the only difference between that and the main out is the master fader itself. The USB record in that version normally connects to REC_OUT, just wired straight across. (I don't like this either, at least not for recording a live performance, because the house mix is usually lousy for recording and vice-versa. But if you're just looking to lay down a demo, then it might be good enough.)
3. AUX sends should be fairly intuitive from looking at the controls. You have a knob at each intersection between channels and AUXes that controls the volume of that channel into that AUX, and you also have a master knob for each AUX. I believe on that board that AUX1 is permanently pre-fade, meaning that the fader at the bottom does not affect it, and AUX2 is switchable between pre-fade and post-fade with a button next to each of its knobs. That's a lot of flexibility for a lot of different uses.
4. The footswitch is for effects on/off without reaching for the mute button on the board. It's not even audio at all.
5. The 1/4" main outs beside the XLR main outs are basically that. Just wired straight across.
6. MONITOR_OUT is meant for a pair of studio monitors instead of headphones. As such, it follows the PHONES signal, but without the internal power amp to actually drive headphones. The LED meter also follows this signal. By default, the main mix goes there, post-fade, but any PFL or AFL button will override it with that signal by itself. Unfortunately that board has no indicators for PFL or AFL, so you have to find and inspect all of those buttons yourself. It technically works to mix a few selected channels that way, but you don't have any control over their relative levels except for the preamp controls at the top of the board that affect everything that that channel goes to.

You've tried almost all of them (you missed the group outs), and a few that aren't, so you might be wondering what they're really for anyway:

1. The 13/14, 15/16, and 2TR_IN (tape input) connectors are inputs of course; they're not going to help you. The USB playback in that version normally connects to 2TR_IN, through an inaccessible mixer with the RCA's that then feeds the volume knob that is the only control between the input and the master fader. (For that reason, I don't like tape inputs or most effects returns. Give me some more channel strips instead, even though they're more expensive.)
2. REC_OUT (tape out) is apparently meant for recording. It's the main mix, pre-fade. Basically, the only difference between that and the main out is the master fader itself. The USB record in that version normally connects to REC_OUT, just wired straight across. (I don't like this either, at least not for recording a live performance, because the house mix is usually lousy for recording and vice-versa. But if you're just looking to lay down a demo, then it might be good enough.)
3. AUX sends should be fairly intuitive from looking at the controls. You have a knob at each intersection between channels and AUXes that controls the volume of that channel into that AUX, and you also have a master knob for each AUX. I believe on that board that AUX1 is permanently pre-fade, meaning that the fader at the bottom does not affect it, and AUX2 is switchable between pre-fade and post-fade with a button next to each of its knobs. That's a lot of flexibility for a lot of different uses.
4. The footswitch is for effects on/off without reaching for the mute button on the board. It's not even audio at all.
5. The 1/4" main outs beside the XLR main outs are basically that. Just wired straight across.
6. MONITOR_OUT is meant for a pair of studio monitors instead of headphones. As such, it follows the PHONES signal, but without the internal power amp to actually drive headphones. The LED meter also follows this signal. By default, the main mix goes there, post-fade, but any PFL or AFL button will override it with that signal by itself. Unfortunately that board has no indicators for PFL or AFL, so you have to find and inspect all of those buttons yourself. It technically works to mix a few selected channels that way, but you don't have any control over their relative levels except for the preamp controls at the top of the board that affect everything that that channel goes to.
7. GROUP_OUT is the post-fade signal from the sub-mix or group faders. Each group appears in stereo on a pair of outputs, so it preserves each channel's pan or balance setting. Groups are handy to make a live mix easier, putting all the vocals in one group and all the instruments in the other, for example. Or you can choose not to send a group to the main mix and use its direct output for something else. That's all done with the assignment buttons below the PFL on each channel, and in the same place next to each group fader to send it to main or not.

1. You said it worked in practice, so it might still be okay. But I can't rule out the possibility of it getting killed somehow in the meantime. I did sound for a wedding once where they handed me a laptop at the last minute to play dance music from...and it turned out to have a bad headphone jack. It never fed the mixer, it didn't appear in Windows' device list like you described, and it didn't even disable the internal speakers. I assume that it worked from the factory, so at some point it must have gotten killed. So that's one possible point of failure.
2. The picture of the jack actually shows a headset jack, not a headphone or mic jack. This is actually a combined connection for stereo headphones using the standard pinout, plus a mono phantom-powered mic by splitting what would normally be the ground connection. (5V through a resistor powers the mic, which returns a signal on its own power line) If that's really what you have and you put a 3-pin plug in it, then you've grounded (muted) the mic input and connected to RCA's to the headphone outputs. I don't know how that worked in practice unless you had a different laptop.
   Also, a dedicated mic input might only have two pins like a mono plug. Same phantom-power scheme there. You might get one channel, but the other gets thrown away. That's not guaranteed though; you might just happen to find a for-real stereo line-input. Consumer and some pro-sumer electronics are just ambiguous that way.

1. You said it worked in practice, so it might still be okay. But I can't rule out the possibility of it getting killed somehow in the meantime. I did sound for a wedding once where they handed me a laptop at the last minute to play dance music from...and it turned out to have a bad headphone jack. It never fed the mixer, it didn't appear in Windows' device list like you described, and it didn't even disable the internal speakers. I assume that it worked from the factory, so at some point it must have gotten killed. So that's one possible point of failure.
2. The picture of the laptop actually shows a headset jack, not a headphone or mic jack. This is actually a combined connection for stereo headphones using the standard pinout, plus a mono phantom-powered mic by splitting what would normally be the ground connection. (5V through a resistor powers the mic, which returns a signal on the same power line) If that's really what you have and you put a 3-pin plug in it, then you've grounded (muted) the mic input and connected the RCA's to the headphone outputs, hence the YouTube video being heard through a channel strip. I don't know how that worked in practice unless you had a different laptop.
   Also, a dedicated mic input might only have two pins like a mono plug. Same phantom-power scheme there. Or it might have three pins like a stereo plug, but still a mono signal with the remaining pin being used exclusively for power. Either way, you might get one channel, but the other gets thrown away. That's not guaranteed though; you might just happen to find a for-real stereo line-input. Consumer and some pro-sumer electronics are just ambiguous that way, and not all manufacturers follow the standard color code either, preferring all black instead, just to add to the confusion.