It’s fine if you accidentally flag a game. They’re reviewed by the administrators and they’re pretty good about realizing it was an accident. I’ve done it before as well. :O
In regards to the missing badge. Click the achievement tab and at the bottom you’ll see a link to report missing achievements. Click it and fill out the report. You need to include a screenshot, but it should be okay as long as everything is visible. You’ll need to show the green plug, your name and some sort of evidence on the screen that you’ve actually done what you’re reporting.
If you want to crop down the picture, simply open it in Paint and drag the box down to the proper size. To do it properly, you’ll need to rotate it 180 degree two times.