The Presidential Succession Act dictates the line of succession. Even if Kirkman was asked to resign, the fact he hadn't submitted his resignation means he was eligible to succeed to the office of President. If he weren't then the Presidency would be vacant meaning a bigger problem than the HUD Secretary assuming the Presidency.
The jail scene was filmed in the Segregation Unit of the Metro Toronto West Detention Centre which has been recently torn down.
This episode takes place in February 2016.