Steam purchase related queries

Sorry for the noob question but I have a confusion. Suppose I buy Counter Strike Global Offensive with my account and install it on my laptop that runs linux and play when I am outside, now I come back home start my windows PC enter steam with my account, now can I play CSGO here as well with that purchase? Or would I need to buy another version for my Windows. Now suppose I go to my cousin's house and wanna play CSGO there, and I enter my account, do I need to re-purchase for that windows as well? Please clear this confusion and really sorry for such a super noob question.

you wouldnt need to purchase it again as long as the playing is done from your account

You'll need to download it again, but normally you shouldn't purchase it again ... unless it's from one of those devs/publishers that insist you may only use one license on one computer and who WILL lock it down if they notice you're using the same game key on 2 PCs at the same time. So be sure to read the EULA when you buy the game.