Question: Just rewatched the series and in the episode where they are operating on President Kirkman to remove the bullet fragments I noticed what I think was a secret service agent in the background of the OR. Is this normal practice for when a president is under anaesthetic or a medical procedure, or was this just because it was after an attempted assassination?
Answer: It would be normal procedure, yes. The Secret Service accompanies the president everywhere, except in very limited circumstances (for example, they don't go into the bathroom with him, or stand in his room while he sleeps). When Bill Clinton had an operation to fix a torn tendon, there was an agent in the operating room throughout the procedure.
Question: If there were two designated survivors (Kirkman and Hookstraten) then why was it decided so quickly that Kirkman would be president instead of Hookstraten?
Answer: The line of succession is VP, Speaker, President Pro Tempore, then the Cabinet Secretaries in the order the department was created. If none is qualified, then the Chief Justice or senior member if no Chief or Chief is ineligible or declines, acts as POTUS until either the House chooses a new Speaker or the Senate chooses a new Pro Tempore, whichever is first.
Answer: While both parties picked a designated survivor, only Kirkman was a cabinet member and in line for succession. Hookstraten was the Majority Whip leader for the House of Representatives and not in line for succession. She would only have become president if something happened to Kirkman.
Question: Why did Lyor, Mike, Brenda, Hookstaten, and Chuck randomly disappear from the series?
Answer: It's a recurring theme on the show. Characters come and go depending on the storyline. General Cochrane, Tyler Richmond, Abe Leonard, etc.
Answer: Yes, it's normal. When President Reagan was shot, they were in the OR with him.