In the first sentence are you trying to say there is a team member position available? If yes - then say “There is a vacant team member position available.” If you are referring to a vacancy left by a current team member then state: “One of our team members left, leaving a vacant position.” The second sentence is correct. The third should read: We did not have internet in the past, so we hand to rush home in time to watch our favorite show.” The forth is confusing out of context. I would state it - We can communicate faster on the internet.

Looks correct. Note: favourite: British spelling. favorite: American spelling.