After you make your picks be sure to hit the "Submit" button at the bottom and then make sure it says your picks are "completed". If it is not complete it will say "incomplete" in red. You can make your picks for next week to see what I'm talking about. You can edit your picks up to 5 minutes before the first game of each week.
As far as the mulligan.. yahoo will Drop 1 score, the lowest score for your season, which for you will automatically be this last week. As long as you don't miss another week this year you should be fine.