Appointment
Ministers are appointed by the president. The chancellor is appointed by the president unilaterally; ministers other than the chancellor are appointed on nomination by the chancellor. In theory, the president can also dismiss ministers: the chancellor can be removed unilaterally; individual ministers other than the chancellor can only be removed on request by the chancellor. The president can also remove, again unilaterally, the entire cabinet at once. In practice, the president does not forcibly dismiss either ministers or cabinets; they resign.
The fact that the cabinet is installed by the president, who is elected directly by the people, and not by parliament is one of the features of the Austrian constitution that impart a semi-presidential appearance to the country.
The appearance is superficial; as a matter of political reality, Austria is a parliamentary republic.
The main reason is that ministers are politically answerable not only to the president but to the legislature as well.
The president is required to dismiss any minister the National Council censures with a motion of no confidence. The National Council can also order to president to dismiss the entire cabinet.
As a result, the president is more or less forced to appoint as chancellor whichever majority leader the legislature preselects.
Austrian presidents gladly accept that their role is that of figureheads. The last time an Austrian president rebuked a parliamentary majority leader was in the context of the installation of the first Schüssel government in 1999; President Thomas Klestil first preemptively announced he would refuse to appoint Jörg Haider,
then in fact refused Chancellor Wolfgang Schüssel's request to appoint Hilmar Kabas and Thomas Prinzhorn.
Even so, motions of no confidence are extremely common; they just never pass. Between 1945 and 2005, opposition parties have submitted no fewer 175 unsuccessful motions of no confidence in order to focus attention and create headlines. There were 65 motions of no confidence in the ten years from 2006 to 2016, i.e. more than six per year. As of 2017, motions of no confidence have been introduced against members of every single cabinet in the history of the Second Austrian Republic.
Unable to govern effectively without active majority support in the legislature, and in fact unable to stay in office without at least its tacit toleration, the cabinet derives not just its actual authority but also its abstract political legitimacy from the National Council.
It is thus customary for the cabinet to resign not just when the ruling coalition breaks down but also whenever a new National Council is elected.
The constitution requires the president to put caretaker ministers in charge of the positions being vacated ("mit der Fortführung der Verwaltung ... betrauen") to prevent disruption of daily operational management of the bureaucracy. In theory, the president could appoint selected departing ministers, state secretaries, or senior civil servants. In practice, the president simply offers to reappoint each departing minister to the office, the ministers accept; the caretaker government is identical to the conventional government preceding it.