It's really easy. It's only about $8 a month (that is for streaming only, no dvds), we have been doing this for years.
Since we have no cable, we watch all of our shows through Netflix. Of course, they are usually a season behind, like I will have to wait a year before seeing the current season of The Office, but it's still nice to have. For current seasons and shows, you can get them through hulu on xbox. but it can be expensive.
As for the technical questions, we just went to our inlaws house for Christmas. My husband didn't bring our xbox, but he was able to sign in to his dads xbox to access our netflix account. You can watch movies/shows online as well.
You don't need to buy additional internet time for xbox netflix streaming. If you are using your sil's computer, you can just go to their website and login to your account and watch it on her computer. If you mean, can you piggyback the internet provider on your sil's provider? Yes, since it is in the same house, it does need to be plugged in the to internet.
You need to be an xbox live gold member, which is about $50 a year. It's actually really easy to set up. Everything is explained here:
http://www.xbox.com/en-US/Live/Partners/Netflix/Home