when a wave-function splits into two paths (such as in a double slit or a mach-zehnder) how does the Schrodinger equation deal with it?

when one of the path is blocked (before/after the slits or anytime prior to reaching the detector) by an opaque obstruction, how is the energy for the wave-function of that path dealt with?

The wave function flows through the system in a manner analogous to a classical wave.
It is usually more helpful to take the phase representation of the wave functions and sum over the available paths.

See the lecture series starting with:

When one path gets blocked, then any particle attempting that path is blocked - it's energy and momentum are absorbed by the blocking material...
Note: the wavefuction does not carry energy, the particle does. The wavefuction is a representation of probability amplitudes of detecting a particle with particular properties in a particular place.

if both slits are open -- the photon(s), even if sent one by one, land up on a different parts of the screen than if only one slit was open.

its as if (at least in extrapolation of some of the interpretations) the wave-function from the other slit is "pushing" the photon around...as if its changing the photon's path....

That interpretation is incorrect. The change in pattern is not consistent with the idea of a wave from "the other slit" pushing the photon around. The wkipedia article you link to does not make that interpretation/extrapolation.

Are you picturing the particle as like a cork being pushed along by a water wave?

I make no claims to have any "correct" explanations of anything.

The paper ZapperZ has shown you (link above) has a nice discussion of the common misconceptions about "wave particle duality" which is the broad umbrella-term for what you are wrestling with.

My personal favorite description comes from Richard Feynman in the lecture series he gave in Auckland NZ ... quite old now but still relevant.

QM waves are not classical waves: they are not waves of anything in the way water waves are waves of water molecules. They are tools for predicting probabilities. They get called "waves" because the math has the same structure as the class of equations in mathematics called "wave equations".

When you get introduced to them, authors like to spell out how weird they are by making comparisons with classical mechanics. This is just why we need quantum mechanics to replace the classical.

So - when you have both slits open - the wave-function in the "other slit" is zero ... because we know the photon went through "this" slit right? Thus the probability that the wave went through the other one is zero.

If we abandon knowledge of where the photon went, i.e. if we only care about the screen, then we work out the probability of detecting it some distance from the slits by working out the amplitude and phase that it would have if it had come from one slit, and the amplitude and phase if it had come from the other slit, and add the two amplitudes together (making a vector sum). The square of this resulting amplitude is the probability of finding a photon at that position.

Here's the Feynman lectures. Watch all of them.
http://vega.org.uk/video/subseries/8/ [Broken]
... and don't worry - it takes everyone a while to wrap their minds around this stuff.