When Toby is talking to the twins in the hospital room and the camera is on him it is quite clear that Huck has a pacifier in his mouth. When the camera changes to look at Huck the pacifier is gone.
The presidential oath is not taken by the Vice President or Speaker upon the 25th Amendment's incapacity provision being invoked. An Acting President is not the same as President. The former merely exercises the powers and duties of the President, without actually holding the office of President.
Leo incorrectly states that a 7700 transponder code is radio failure. 7500 is the code for hijack (which is correctly referenced), 7600 is the code for radio failure, 7700 is emergency (which is code the President asks Leo about).
The justice tells Speaker Walken to put his right hand on the Bible and raise his left hand to take the oath of office. This is backwards. He should place his left hand on the Bible and raise his right hand.
When the Speaker of the House resigns from his office in order to be sworn in as President, Leo tells him he cannot "just go back" but has to be re-elected in two years. This is not true, because the Speaker does not have to be a current member of congress (so the house can just elect citizen Walken to be their speaker). Also because congressman Walken resigned his, congressional district can hold a special election, so he can be back in as little as six months as representative and within two weeks as Speaker.
Josh recommends to Leo that the Treasury Department suspend trading on the stock exchange. The Securities and Exchange Commission is an independent agency that regulates the stock exchanges, not the Treasury Department. Additionally, any closures of an exchange similar to after 9/11 would be conducted by the exchange officials - not US Government.