It worked for me- you just have to click inside the window. It comes down to how you handle locking the mouse cursor in the game, but I think it'll always be a bit janky.
Also, I'm not sure about the white bars, but I assume it has to do with the aspect ratio you're working with- when you upload a Unity game, you might need to open the index.html file and change the dimensions written in there like "height="600px" width="960px", or whatever your dimensions are.
I'm really bad at making the sizes work, so I can't really help with the actual numbers you should use, but I think that's the problem.