quantum mechanics - Queries of proof of adiabatic theorem in QM

I have a few questions regarding the proof of the adiabatic theorem in the book "Introduction to Quantum Mechanics" by Griffiths:

The assumptions are that if the Hamiltonian changes with time then the eigenfunctions and eigenvalues themselves are time-dependent: $$H(t)\psi_{n}(t) = E_{n} \psi_{n}(t).$$

1. Firstly, how would we know that the spectrum remains discrete so as to write it in this way?

He then states that the eigenfunctions form an orthonormal set which is complete, this I understand comes from the postulates of QM. But then he states that the general solution to the time-dependent Schrodinger equation can be expresses as a linear combination of them: $$\Psi(t) = \sum_{n} c_{n}(t) \psi_{n}(t)e^{i \theta_{n}(t)} ~~~\text{where }~~\theta_{n}(t) := - \frac{1}{\hbar}\int_{0}^{t}E_{n}(t')dt'$$

2. How does he know that this is the form of the phase factor (he uses this later in the proof)?