How would I double up the wires? I untwist the speaker cable and split it?

No, never untwist stranded wire; that makes it likely to break. Also, Ethernet jacks expect solid wire. You need to splice 2 wires onto the speaker wire. The easiest way is to get some “Wago” connectors — they're very easy to use and accept a wide range of sizes.

You say to not really care about thin wire, but is it a fire hazard?

If speaker wiring were readily able to be a fire hazard if done wrong, there'd be a lot more warnings accompanying it. Unless you're hooking up a subwoofer driver (not your case, since you're using a powered subwoofer, I assume), there just isn't that much average power in music, and for thermal considerations, it's average power that matters.

Finally, can I use any speaker cable to connect to the Ethernet splitter? Does the AWG matter?

For the splitter that you hook up on to the plug side, there are all sorts of different models, but a lot of them will have screw-clamp terminals that can take fairly thick wires — maybe 14 or 16 AWG maximum, depending on what you pick. With the doubled (paralleled) wiring I suggested, you'll also need a couple of splice connectors to combine 4 to the 2 wires to the speaker.

Inside of the box, for the keystone jack, most will be “punch down” terminations which require fairly small solid wire — likely at most 22 gauge (commonly known as "hookup wire"). (Look for “toolless” jacks for the easiest installation, or use a tiny screwdriver to push the wire in if you want to save money.)

Oh, and one more thing — after setting this up, before hooking it up to your amplifier and speakers, use a continuity tester (get a cheap multimeter if you haven't got one) to verify that none of the 4 wires are accidentally shorted, wired to each other instead of separate. That way you avoid blowing a fuse on your amplifier. You can also use it to make sure you've got the left and right correct and not got a phase reversal.