1. The concept of majority and minority carriers is applicable or to (or meaningful for) situations where semiconductor is not far from equilibrium (quasi-neutrality) - where majority carrier concentration is (approximately) equal to the (equivalent) doping concentration, and minority carrier concentration is much lower than doping concentration - and where semiconductor is not affected by the field from other elements (like gate).
MOSFET is a unipolar device - i.e. its operating principle is based on carrier transport form start (source) to end point (drain) by one carrier type, and the other type carrier is not essential for the current flow.
The most important element in MOSFET is (inversion) channel - which is usually formed by the gate, and large voltage is applied to it.
Nobody calls the carriers in the channel majority or minority - the situation there is very far from equilibrium.
2. In pMOSFET, current inside the semiconductor is carried by holes, but these holes are recombined at the interface contact/source-drain with electrons - so that the current outside the semiconductor (interconnects routing / metallization) is carried by electrons.
so, current flows continuously (in transient, switching condition) across nMOS and pMOS devices in an inverter - there is no current interruption.
3. yes, and they flow through the channel, or inversion layer.