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The principle for recording is the same. You can watch my video on recording from mixer to laptop. Get the signal from the main output of your mixer into your audio interface, from your interface, get the signal into your computer. You can use any audio editing software to record as long as you can get the signal into the computer.
Noise gate is like a filter that is applied on a particular channel of sound to exclude bleeding sounds which is not originally intended to be in that particular channel. Take for instance, the mic on your drum kit, you will need to apply a noise gate to each of the mics because those mics will be picking sounds from the surrounding kit. For instance, the snare mic can also pic sound from the kick or the hi-hat. So you can apply a noise gate to filter out the sound of the hi hat if the intention is not to include it in that channel so as to have a clean feed into the mix.
If you are using an analogue mixer, you'd have to get an external gate, except your mixer comes with it in built, but it'll be stated there on the mixer. But most (if not all) digital mixers have gates