There's not much yet in terms of FAQ or support questions, I'm sure this FAQ will grow with time, but for now, here's what I got :
Q: Can I do 'this' or 'that'?
A: Surprisingly, I received a few questions on how to do things that are already supported, so I would suggest the first thing to do is to check your extension settings.
Click on the Beyond20 icon in your browser's toolbar, if you are on a D&D Beyond character sheet, it will pop up the quick settings and character sheet specific settings. If you are on the Roll20 VTT page, it will pop up the quick settings and Roll20 specific settings. In both cases, you can click the More options button to access the full extension options.
You can also go to your browser's Extensions page and open the options for Beyond20 that way.
Q: All I see in Roll20 is an empty message, how can I fix it?
A: That's because the game in Roll20 is set up to use a character sheet template other than "D&D 5e By Roll20".
Beyond20 uses display templates from that character sheet template to make the rolls, so if it's not the one configured in the Roll20 game, then it won't work.
To fix the issue, simply go to the Roll20 game settings and select "D&D 5e By Roll20" for the character sheet.
If you cannot do that (you're not the DM) or you don't wish to do it (you prefer another template), then go to the Beyond20 options and select "Other templates" in the "Character sheet templates" option.
Q: HP from D&D Beyond doesn't synchronize to Roll20. Am I doing something wrong ?
A: Make sure you have at least version 0.0.9. Make sure the HP syncing is enabled for that character (click the Beyond20 icon in the toolbar when in the character sheet) and make sure the names match between the character in Roll20 and D&D Beyond (non case sensitive).
Q: I have the <Class feature/Feat> (Sharpshooter Feat for example) but I don't see the option to use it from the popup options even though I know it's supported.
A: Beyond20 doesn't have access to all of your features, but it will save the list when it finds it, so if you've changed your class features (leveled up) or Feats (taken a new one), then you need to click on the "Features & Traits" tab to have the list displayed on the page for Beyond20 to pick it up and save it. After doing that, the option to use the feat should become available to you (assuming it is a supported feature that does have an option).