1st of all, Windows 7 is not a computer, it's an operating system. But if you are looking to get mumble, you can go to the
mumble home page and click the banner off to the right that says "Mumble Client 1.2.3a", and it should open a mini window. On this mini window, it shows the options of mumble where you can get a skin(it should say "with Mumble.com Skin" underneath it), and where you can't get a skin. I recommend the skin allowing one if you like personalizing stuff, but if you don't, you can go ahead and click the bottom one. After that it should download the ile onto you computer. If you are not using Google Chrome, go down into your windows search bar in the start menu, and search "Downloads". Click on the one with the translucent looking folder icon, and select mumble-1.2.3a_plus_MumbleComSkin.exe if you downloaded the one with the skin, but if you didn't, it should just be mumble-1.2.3a.exe. When you locate either one of those files, double click on it and it will run you through the steps and install mumble on your computer. If you are using Google Chrome, all you need to do is go down to the downloads bar and run the file and you should be all set.
As to your questions about the headset, you will only need it if your computer does not have a built in microphone. Most computers do, so you should be good to go, but if not, you will indeed need a headset with a mike.