Oh, sorry, I thought you'd gotten it figured out. The cables question is pretty simple, but I don't have any experience recording that sort of sound (especially from a distance in a large space), so if no one else chimes in with comments about that, I guess you'll have to do some practice to get the settings straight. It would probably also be useful to do a web search for an "audio"- or A/V-type forum (rather than a "tech"/computer-oriented one like Anandtech) where you'd be more likely to find help as far as microphone placement, mixing levels, etc, etc are concerned. The only real advice I can give is to turn the mixer's output ("fader") control down very low when you first start testing the setup. The higher the output level from the mixer, the more likely you are to get clipping/distortion in the recording. Basically, you eventually want that level set only as high as necessary to get a good recording. As for the cables...
Are you set in terms of connecting the mics to the mixer? I'm assuming you already have mics and mic cables, and that the cables have XLR connectors that you'll just plug into the jacks along the very top of the panel of the mixer.
For the output, you have options, but I frankly have
no idea which would give you the best sound quality, or if it it would make a significant difference which you use. If you're anxious to get the best recording possible, it might be worth buying both cables since they're not expensive, trying them separately during test runs, and comparing the recordings.
The simplest would be to connect the stereo-jack "AUX" output on the mixer (#23 on the diagram on page 3 of the mixer's manual) to your computer's audio input using a cable with 1/4" stereo TRS plug on one end and a 3.5 mm (aka "1/8-inch") stereo plug on the other. But you could also use use the last of the cables I've linked below to connect the mixer's separate left and right "main" outputs (#25 on the mixer's panel diagram) to a single 3.5 mm stereo plug. In theory, the shorter the cable from the mixer to the laptop is, the better sound quality you're likely to get, so if possible, I'd go with the 3 foot cable, but "probably" anything 10 feet or shorter would be OK.
Here are a few links to products on Amazon as examples. I can't speak to their quality from personal experience and it looks like both have good average ratings over many reviews, but while appearances can be deceiving, the UGREEN cable at least looks a bit heavier/better constructed than the Hosa cable, for whatever that's worth. I didn't check myself, but if there's anything in the product specs, questions and answers, or reviews about the gauge of the actual conductors inside the cables, lower gauge/heavier wire would be better on principle, as would any overall shielding of the cable.
1/4" stereo plug to 3.5 mm stereo plug:
UGREEN 3.5mm 1/8" Male to 6.35mm 1/4" Male TRS Stereo Audio Cable (3 ft) (with longer options on the same page)
or
Hosa CMS-110 3.5 mm TRS to 1/4 inch TRS Stereo Interconnect Cable, 10 feet (the same basic type of cable as the UGREEN)
Separate mono 1/4" plugs to a 3.5 mm stereo plug:
Hosa CMP-153 3.5 mm TRS to Dual 1/4 inch TS Stereo Breakout Cable, 3 feet
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