Why Claire Will Get Away With Doug's Murder
Murdering a man in the Oval Office might be the most brazen crime either Frank or Claire has ever committed - and yet it's pretty clear that she's going to get away with it. Colonel Flora is already in custody, and after entering her office Doug handed Claire a list of all the assassination co-conspirators, which she can use to take down her enemies - Annette Shepherd included. Claire deliberately left the letter opener on the desk and then pushed the distraught Doug to the edge (a dangerous gambit, to be sure, but one that paid off). The wound on her neck is proof that Doug was there to kill her, so she can simply say that she stabbed him in self-defense. It's the silence of a dead man who was believed to be mentally unstable and had already been previously charged with murder against the word of the President of the United States, and a pregnant woman no less.
Claire is also now in possession of Frank's audio diary (which Doug would never entrust to anyone else, because he knows that some of its contents could destroy Frank's name), so all her other secrets will remain hidden as well. She's won... for now.
Will Claire Still Launch A Nuclear Strike?
With Doug dead and the threat of what he knows (as well as the threat of Frank's audio diary) gone, Claire no longer really needs the distraction of the nuclear crisis. In an aside earlier in the episode, she admitted that she really didn't want to have to go down the route of a nuclear attack, so it's likely that she would back down from the threatened strike on Syria - perhaps revealing that the ICO terrorist leader is dead, and even making moves to heal the relationship between the U.S. and Russia. Open war has never been Claire's goal, and it's not hard to imagine that she would find a way to stop the current situation from snowballing.
Of course, that's little comfort now that we know just how far Claire is prepared to go in order to protect herself. Even if she doesn't launch a nuclear strike this time, a situation could arise in the future where she pulls the same move - and follows through on it. Doug Stamper has handed her the evidence she needs to take out her biggest political enemies, which may keep Claire safe for a while, but we wouldn't put it past her to initiate a nuclear war if pushed to the edge again.
Who Is The Father of Claire's Baby?
As far as America is concerned, the widowed President Hale is carrying her late husband's baby... but the audience knows that things aren't quite that simple. In a conversation with her younger self, Claire suggests that there's a "50-50 chance" that she's carrying a "demon seed," which strongly implies that she doesn't know whether it's Frank's baby or Tom Yates' baby. Claire definitely seemed to be more sexually active with Tom in season 5 than with her husband, but that doesn't completely preclude the possibility of the baby being Frank's.
With this being the series finale, it seems that this particular question will be left unanswered.