maybe if you provided a sample of what sounds you are trying to control, i could suggest something. Truth is not all noise is the same, some can be filtered out, some have to be muted, other techniques include adjusting phase and interpolating. You may be able to address some of it by having the microphone lower the noise when you are not talking. If the noise is too loud, then you may have to address this from before the mic, as it might be more trouble than it's worth to clean it. In some cases, removing noise can be like un mixing coffee, in the sense that the noise frequency and the frequencies that you want to keep, like game and voice share the same data - so it's like mixing white and red paint, then trying to un mix it to make it white again. - and since i'm on that analogy, all the layers of sound from your environment are mixed through the mic. - so adjust the mix before it gets to the mic ( if that makes any sense lol )
ps - if you want to get scientific about it, there are techniques that are used to phase out sounds - such as how it was handled in the printing press. I'm not going to go into detail, but if you look up how they handled noise control in old school printing presses - it might give you an idea of how to do that creatively at home.
a brief explanation of how it works, the sound of the environment is fed back to itself from a speaker with the phase inverted, this cancels the audio and starts to become invisible to the microphone. They key word too look up and learn more about this topic is "Phase Cancellation"