Firstly, there are a few issues with a time-dependent potential, $V(x,t)$. Namely, if we apply Noether's theorem, the conservation of energy may not apply. Specifically, if under a translation,
$$t\to t +t'$$
the Lagrangian $\mathcal{L}=T-V(x,t)$ changes by no more than a total derivative, then conservation of energy will apply, but this resricts the possible $V(x,t)$, depending on the system.
We often treat each Schrödinger equation case by case, as a certain system may lend itself to a different approach, e.g. the harmonic oscillator is easily solved by employing the formalism of creation and annihilation operators. If we consider a time-dependent potential, the equation is generally given by,
$$i\hbar\frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial \mathbf{x}^2} + V(\mathbf{x},t)\psi$$
Depending on $V$, the Laplace or Fourier transform may be employed. Another approach, as mentioned by Jonas, is perturbation theory, whereby we approximate the system as a simpler system, and compute higher order approximations to the fully perturbed system.
Example
As an example, consider the case $V(x,t)=\delta(t)$, in which case the Schrödinger equation becomes,
$$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + \delta(t)\psi$$
We can take the Fourier transform with respect to $t$, rather than $x$, to enter angular frequency space:
$$-\hbar\omega \, \Psi(\omega,x)=-\frac{\hbar^2}{2m}\Psi''(\omega,x) + \psi(0,x)$$
which, if the initial conditions are known, is a potentially simple second order differential equation, which one can then apply the inverse Fourier transform to the solution.
I don't know whether Schrödinger proved or guessed the equation with his name, but this equation can be derived similarly with the diffusion equation - see Gordon Baym, "Quantum Mechanics".
However, differently from the diffusion equation, the diffusion coefficient in the Schrodinger equation is imaginary. That tells us that we have to separate the Schrödinger equation into two, one equating the real parts of the two sides, and one equating the imaginary parts.
The meaning of this imaginary diffusion coefficient is therefore that the wave-function is complex, or, in other words, it has an absolute value and a phase, like the electromagnetic wave.
Best Answer
You're looking for some form of differential operator that will take your plane wave $$\Psi(x,t) = \Psi(0,0)\exp\left[\frac{2\pi ix}{\lambda}-i\omega t\right]$$ and will return $p^2\Psi(x,t)$. As you've noticed, you can apply a space derivative to get $$ -ih\frac{\partial}{\partial x}\Psi(x,t) = p\Psi(x,t). $$ To get another $p$, you can apply the space derivative again to get $$ \left(-ih\frac{\partial}{\partial x}\right)^2\Psi(x,t) = -ih\frac{\partial}{\partial x}p\Psi(x,t)=p^2\Psi(x,t), $$ but as you note you could also square the whole thing, to get $$ \left(-ih\frac{\partial}{\partial x}\Psi(x,t)\right)^2 = p^2\Psi(x,t)^2. $$ The reason we take the former version and not the latter is that the second form introduces an extra factor of $\Psi(x,t)$, which would make the wave equation nonlinear and introduce all sorts of terrible, horrible nightmares.
In the end, the argument is heuristic, and you shouldn't read all that much into it. The Schrödinger equation (particularly once you introduce a potential) is a postulate in its own right and cannot be derived flawlessly from first principles. You can only offer some justification for its form, which is what you're trying to do.
In detail, the argument you're making is of this form: if we believe the de Broglie relation $p=h/\lambda$ and if we believe the Planck relation $E=\hbar\omega$, both of which give us the particle properties in terms of the wave properties, and if the matter wave is a plane wave of the form $\Psi(x,t) = \Psi(0,0)\exp\left[\frac{2\pi ix}{\lambda}-i\omega t\right]$, then a wave equation of the form $$ i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} $$ reproduces the correct classical relationship between the energy and the momentum, $E=p^2/2m$.
That's all that the argument can claim. In particular,
The miracle, of course, is that when you disregard all three of those objections, you introduce the potential as a pretty big kludge, and you look for the non-plane-wave solutions for some complicated problem like the hydrogen atom, you get something that matches pretty exactly with experiment. This is the real justification for the Schrödinger equation: an a posteriori justification based on its success in replicating experimental results, instead of an a priori justification based on some over-arching set of first principles.