I did not have a chance to write a detailed post earlier, but I will try to answer your questions better now.
On #2, the gain knob at the top of the channel strip is for the mic pre-amp. This takes the weak electrical signal from your microphones and amplifies it to professional audio line-level inside the mixer. From there, you have your EQ, AUXes, and eventually, the channel fader. The insert I/O jack is either before or after the EQ -- I can't remember which and some mixers even offer a choice via a DIP switch or solder points you can modify.
The reason this is important is you may have a compressor or gate connected via the insert I/O jack. That receives a pre-fader signal. So do pre-fader AUXes that you may use for stage monitors. If you engage the SOLO button and the meter/headphones are in PFL (pre-fader listen) mode, you can see the signal level at this point. This is how you should adjust the pre-amp gain. On an analog board with 20dB of headroom like the ZED, you probably want your peaks in the range of +0 to +6dB. Just into the yellow. You do not want the red clip light to come on -- that means the signal is being distorted and something needs to be turned down.
The fader is just another ordinary op-amp that controls the "volume" of the signal going to the main mix, as well as post-fader AUXes which are typically used for FX. That way the signal going to the FX processor is adjusted along with the one used for the main mix, so you don't change the wet/dry ratio of the signal every time you adjust the fader.
How you adjust the channel strip faders is totally up to you. As long as you have some head-room to increase the "volume" for soft singers, and some room to go down for loud ones, you are basically fine. Don't over-think this, just do what sounds right.
On the main fader, this is typically the master volume control for your audio system. Since you have a ton of head-room in your amps and speakers, you might run it really low. Alternatively, you might turn your amp knobs back so you can send a hotter signal from the mixer and get more use out of its LED meters. Finally, you could be technically correct by adjusting the dials on the amp so it clips at the same level as the output of the mixer clips. Being technically correct is not always the easiest or best thing to do though, especially with karaoke, which is not at all like mixing for a band.
As far as your #1 question about the computer volume goes, here is my take on that. The volume control in compuhost is a digital gain attenuator, and it should basically always be set to 100%. If you start using EQ or something in your karaoke program (I am not familiar with compuhost) then you may need to give yourself some head-room to avoid digital clipping.
The main output of your computer, the windows volume control or whatever, is also a digital gain attenuator, so generally you want that at 100% also. As long as you do not hear distortion that might be caused by over-driving the laptop sound card's op-amp, do not worry about turning it down. If you do hear analog clipping, reduce the windows volume control.
Finally, when you hook in to the mixer, use the PFL meter to adjust it so the musical peaks are somewhere around +6dB. Then use the fader to get the appropriate volume of music into the mix. If you need music in your monitors and you use pre-fader AUX sends for them, adjust those knobs as needed also.
One last thing to mention, your mixer probably has a lot of head-room, and +6dB is a conservative number for most analog audio systems these days. It is about right for recording, and it gives you some head-room for EQ. If you are doing things like adding 15dB of gain to a particular signal band using EQ, though, you should remember that when you think of how much head-room you have between that +6dB and the point where things start to clip, probably around +20dB.