1) Zuck, with his 57% voting proxy, has absolute control over the company. He doesn't need to consult anybody for his decisions, even Sandberg or the board.
2) With that said, he made sure they voted. The board still needs to vote for it to be legal, but if they were to vote it down, Zuck could simply remove the dissident board members with his 57% share.
3) This acquisition was defensive, not offensive, in nature. Yes, Facebook could have created its own filters, but it was worried that Instagram could become THE social network for sharing images. That would be bad for Facebook, since that's it's the undisputed leader in that space. Zuck knew it would be more expensive to leave Instagram independent than to take it out now for a high price.
4) No matter who acquired Instagram, there were going to be conflicts of interest. Twitter's CoI is actually worse -- it's Executive Chairman, Jack Dorsey, was an angel investor in Instagram. This is why Zuck doesn't make investments, I suspect.