When Robin left the room at the end of the previous episode, Lily and Robin's sister Katie were there. This episode opens only moments later in the same room, and yet Katie has vanished and in fact doesn't appear again, not even at the wedding.
Ted says in "Last Cigarette Ever" (5.11) that Lily's last cigarette ever was when she decided to start trying to get pregnant. Thus Ted should not have expected Lily to still be smoking after she had already had a baby.
Marshall states Lily would have put the butt in the plant to let it degrade and feed the plant. Cigerette butts are made of plastic fibers that do not degrade and contain chemicals from filtering the smoke that would have hurt the plant.
Lilly's wrong about her daughter not being able to run for President if she's born in Rome. A child born to an American citizen anywhere in the world is a natural born American citizen. George Romney was born in Mexico to American parents. John McCain's parents were in the Panama Canal Zone when he was born. The Senator Ted Cruz was born in Canada to an American mother. All these people were natural born Americans. And any child born to Lilly would be an American as well.
In this episode the characters say they only know a few things about Robin's mom when early on in the show they talk about Robin's mom several times like when they said she was facing some severe health issues in the episode "Perfect Week"
The daisy was obviously not a real plant.
Lily tells Marshal if their baby is born outside the US, "she can never be President." This is untrue, as the Constitution says only that the person must be a "natural-born citizen" which means they cannot be someone requiring "nationalization." Any child of 2 US citizens would be a "natural-born" citizen.
For someone who claims to be an accomplished fencer, the Captain does not assume a correct stance to start fencing (the "en garde" position).
The Captain claims to have won a gold medal in fencing. No American has won a gold medal in Olympic fencing.