In the USA, there won't be any major anti-trust initiatives while Trump is President, so the main worry that Facebook has to think about is whether the European Union might take action. But in Europe they've so far been willing to regulate American tech giants, without attempting to break them up. There's also little precedent for the European Union actually breaking apart a company that is headquartered in the USA. I can foresee some fines for Facebook, but nothing they can't manage. They'd probably be happy to pay some fines, so long as they get to keep their monopoly.
I wouldn't be so sure about that. There certainly won't be any antitrust suits against his Manhattan banking buddies, but Trump has already mused about breaking up Amazon and I doubt he's a big fan of the other big tech companies either (since they certainly aren't fond of him). I would rate a DOJ action against Facebook in this administration as "unlikely, but far from impossible".
I don't see Facebook being much different though. Zuckerberg's politics probably aren't all that different from Bezos', and Facebook's control over the news could easily catch Trump's attention in a bad way, especially if they really follow through with cracking down on "fake news", which (let's be honest), bipartisan problem though it might be, helped/helps Trump's side a lot more than the other one.
The Washington Post is one of the leading journalism sources in the investigations of Trump connections to Russia, Facebook just puts everyone into an echo chamber. Facebook does not meet the minimal standard of being a threat to Trump in some way that I implied but did not spell out. There is minimal chance of Facebook cracking down on fake news - historically Facebook showed how much of spine they had when they crumbled at the first sign of criticism from conservatives about Facebook's program to use real people to curate stories and the employees were spiking stories aimed at conservatives that were fake.
