How to interpret an inner product between two non-degenerate eigenstates of time-dependent Hamiltonian?

There are three questions I would like to ask:

This finally enables us to rewrite $(5.6.8)$ as
$$
dot{c}_{m}(t)=-c_{m}(t)langle m ; t|left[frac{partial}{partial t}|m ; trangleright]-sum_{n} c_{n}(t) e^{ileft(theta_{n}-theta_{m}right)} frac{langle m; t|dot{H}| n ; trangle}{E_{n}-E_{m}}
$$
which is a formal solution to the general time-dependent problem. Equation (5.6.10) demonstrates that as time goes on, states with $n neq m$ will mix with $|m ; trangle$ because of the time-dependence of the Hamiltonian $H,$ by virtue of the second term.

(Reference (A) to above sentence: Modern Quantum Mechanics, J. J. Sakurai, pp347)

Along with above sentence Reference (B): Non Adiabatic Coupling Term in Born Oppenheimer Approximation

Questions:

1. How to interpret this term:
   $langle m ; t|dot{H}| n ; trangle$ ?

2. Why and how does Sakurai interprets the second term (summation) as states are mixing as time goes on?

3. As in reference (B), the meaning of an adiabatic process is given to be a process in which the "system does not pass through something else". How does these two statements (from reference (A) of mixing and above) interpret the same?