1) Why do you believe that instantaneous probability densities are not meaningful?
2) Essentially any non-stationary state for which you need to compute time-dependent wavefunctions: e.g. chemical reaction dynamics, particle scattering, etc.
3) Yes, the time dependant Schrödinger equation applies to isolated systems.
4) By definition energy is conserved in an isolated system. Moreover, the Schrödinger equation conserves energy because the generator of time translations is the Hamiltonian and this commutes with itself $[H,H]=0$, i.e. energy is conserved. For isolated systems, the Hamiltonian is time-independent (explicitly) and the time-dependent wavefunction $\Psi$ has the well-known form $\Psi = \Phi e^{-iEt/\hbar}$, with $E$ the energy of the isolated system.
5) I do not understand the question.
Any equation claiming to describe fundamental process has to have time. We know that in nature system changes with time. They are not static. For example, absorption of a photon by a hydrogen atom is a dynamic process and we need to have a theory which can predict this dynamical process in time. Since Schrodinger equation is supposed to be an equation describing nature, time needs to get involved somewhere.
The question then is why did Schrodinger choose this particular equation? There is no rigorous argument to that. If you read Schrodinger's original paper, you will see that Schrodinger used Hamilton Jacobi Equation, $H(q,\frac{\partial S}{\partial q})-\frac{\partial S}{\partial t}=0$. He replaced action $S$ by, $$\psi=e^{KS},$$
where $K$ is some constant with the dimension of action (seems familiar?). He then postulates that integral of the left-hand side of above equation should be extremized. This leads to Schrodinger Equation. The factor of time derivative in Schrodinger equation comes from the term $\frac{\partial S}{\partial t}$ in Hamilton Jacobi Equation.
Above procedure may seem, and is, ad-hoc. But he was able to calculate energy spectrum of Hydrogen atom successfully with it. He also wrote a paper in which he sort of gives motivation for this procedure. He used the geometrical formulation of Hamilton Jacobi Equation and argued that quantum physics differ from classical in the same way wave optics differ from geometric optics.
In short, we require wave function to have time dependence if it has to describe nature. The way this dependence was introduced by Schrodinger is little bit shaky but not entirely bogus. I would suggest you to read his original papers if you want to understand his motivation.
Best Answer
It is important to understand how the eigenfunctions are defined. In particular, here we talk about the eigenfunctions of the hamiltionian, i. e. functions that satisfy
$$\hat{H} \psi(x) = E \psi(x) , $$
where $\psi(x)$ is an eigenfunction (function of space only) and $E$ is its corresponding eigenvalue. In case of the hamiltonian the eigenvalue is also the energy of its associated quantum state.
Now, the total eigenfunction is a function of space AND time. The form of these full eigenfunctions is:
$$\psi(x, t) = \psi(x) \exp{(-iEt)}, $$
and the probability density is $P(x, t) =\psi(x, t) \psi^* (x, t) $.
Now you can see that if one takes a single eigenfunction, then the time-dependence (the exponential part) cancels with its complex conjugate in the probability density.
If one takes a linear combination, e.g. $\psi_1 + \psi_2$, then the probability density is:
$$P(x, t) = \left( \psi_1(x)\exp{(-iE_1 t) } + \psi_2(x)\exp{(-iE_2 t) } \right) \left( \psi_1(x)\exp{(-iE_1 t) } + \psi_2(x)\exp{(-iE_2 t)} \right)^*. $$ Taking the complex conjugate one obtains
$$P(x, t) = \left( \psi_1(x)\exp{(-iE_1 t)} + \psi_2(x)\exp{(-iE_2 t)} \right) \left( \psi_1(x)\exp{(iE_1 t)} + \psi_2(x)\exp{(iE_2 t)} \right). $$
Now there is no reason why the above should be time independent. The exponentials with $E_1$ do not cancel exponentials with $E_2$, therefore one expects mixing terms such as $\exp{i(E_1-E_2)t}$ in the final probability density, and thus it will no longer be time independent.