You'll never get wood to pass an engineers peace of mind.
I don't really know what you're talking about in terms of your "washers".
Yes you do have to bolt your box down, how else were you intending to keep it where it is? Tie it down with shoelaces?
No, the plastic isn't very strong, that's why you have to spread the load somehow, you don't want to just have the bolts go through the plastic alone because in the event of an accident it'll just rip the plastic to shreads. To solve that problem I got a piece of aluminium and put it in the box, and passed the bolts through those. That way, the plastic box is sandwiched between the plate and the boot floor, which makes it a hell of a lot stronger.
Then to mount the battery brace, I passed it through the plate too. So it isn't the plastic box holding the battery down, it's the aluminium plate.