To answer the title:
Does the intrinsic spin of electrons change when we excite electrons for hybridisation?
In physics, intrinsic spin is the intrinsic angular momentum = to 1/2 that characterizes the electron, and means it is invariant.
If you mean the projection of the spin, (which can be + or -1/2) , it depends on the particular interaction's angular momenta; in your case the angular momentum characterizing the energy level, the projection of the spin of the electron can change. It is angular momentum that has to be conserved.
I am interested in knowing how exactly do they "jump" from one level to another.
The electrons in energy levels of atoms or molecules are in orbitals, not orbits, i.e. probability loci. To get a feeling for orbitals see this calculation for the hydrogen atom. To change energy levels energy has to be supplied , a photon with the energy difference from level to level, and the photon spin will also be involved in the probability of the given transition. This could be a virtual photon, and one has to study quantum mechanics to understand virtual photons (see the second page in this link).
In general it is quantum mechanical probabilities that control transitions in atoms and molecules.
Do the electrons move as a particle or as a wave through energy levels (because if they move as a particle, the distance and space between the electron shells must be covered) because electrons exhibit wave-particle duality?
The "wave" in wave particle duality is a probability wave, how probable it is to find the electron when measured at (x,y,z,t). One is in the realm of quantum mechanics, not classical physics with orbits.