1) A litigant can always appeal a trial judge's grant of dismissal. In this case, I would be surprised if they didn't since the judge may have overstepped his bounds by making factual findings that should be left up to a jury. I'm not saying an appeal court will reverse the trial judge, just that an appeal seems sensible in this case.
2) In this case, the plaintiff is a private person, as opposed to a public figure (i.e., celebrity or politician). He only needs to prove that the defendant (WaPo) published a defamatory statement negligently. Obviously that means he has to prove (a) that WaPo published something that is false (not just a matter of opinion) and (b) did so negligently (a reasonable person standard). He does not need to prove WaPo published the stories with a reckless disregard for the truth or malice. That standard only applies to public figures.
3) The fact that the suit is seeking 250 million should have no bearing on whether the case goes forward or not. Damages are determined after a finding of liability. I think the 250 million was just something to bring more attention to the case in order to vindicate the plaintiff in the court of public opinion, if not the court of law.
I'm not predicting he ultimately wins, just attempting to correct some common misunderstandings.