Mic picking up too much sound

There are multiple ways to adjust the sensitivity of your mic, both in-game and within your OS. A quick internet search should give you some options to try.

Personally, I prefer press-to-talk, but I'm a bit OCD, so that is to be expected.

Put yourself on push-to-talk.

Or take apart your keyboard/mouse and pad every key/button with some of that thin foam that logitech wraps their speakers in. You can also find something suitable at various hobby/crafts stores. You'll also need some glue that won't react to the foam or the surface you're attaching it to.

I cough frequently, so push-to-talk is an absolutely MUST for me.

You could reduce the microphone levels/mic boost in Windows and Skype and try to find a sweet spot. I don't know what microphone your headset uses but it could also be an omnidirectional one which picks up noise from all directions. My old/shitty Logitech one had that and it picked up my keyboard clicks as well as cars driving outside the house.

Lower the mic gain. You can search for "windows boost mic gain" to find some system controls.
I never use push to talk and I'm good. I'm also using a 10 USD headset and my pals listen to me just fine.

If you want to invest, there are headsets with noise cancelling. You should research a bit on wikipedia about that too (it's not magic).