If the superpositions of a wave function are not needed because only one of the superpositions is allowed, what happens to the eigenvalues of the "null" superpositions?
Is the energy transferred elsewhere such as alternate universes, supporting theories such as higher dimensions and inflation theory?
Could these collapses come from string harmonics, therefore determining our universe and the others plausibly around us?
The loss of superposed states from a quantum wave function works a bit more simply than that, although not it's not particularly comprehensible. It's just what is seen experimentally.
Whenever alternative quantum states disappear, the historically recorded history of the particle makes it look as though the particle simply followed or was in that state all along. Thus for example, if the way you found the particle implies that it moved a long a certain path, then the history of that path becomes quite real, and the histories of any other possible paths simply cease to exist. They transfer no energy, they run no races, they leave no traces. They just never... were.
Mulling over such issues is one reason why physicists as great as Louis de Broglie and more recently John Stewart Bell (of Bell's Inequality fame) believed in a curious idea called pilot wave theory. (For the record, very few English-speaking physicists accept this idea anymore. I do not accept it because I feel it flatly contradicts Richard Feynman's QED theory.)
In the pilot wave perspective, there really is only one particle, and there really is only one history. A quite separate and nicely mysterious associated wave then guides (or "pilots") this real, singular particle through the various hoops that make it behave as if it were part of a quantum wave.
In any event, it's just this kind of perplexing "Macavity wasn't there!" behavior of the alternative quantum states that can drive people a bit nuts when they think too hard about quantum mechanics. It is also the driving effect behind the development of popular ideas such as the Many-Worlds Interpretation (MWI) of quantum mechanics.
You do not understand what does Von Neumann’s state vector (wave function) reduction mean. It is not about other eigenvalues which are not needed. It is about something else: after a quantum measurement the coherent superposition is destroyed. If the observer measured one of eigenvalues, then (the same) observer henceforth lives with only this possibility and hasn’t access to any other. The question whether alternatives exist is, as of present development of science, a metaphysical one rather than physical.
The question about energy is ill-related to the problem of reduction. If we measure the system’s full energy (using the $\hat H$ operator) when it was in a superposed quantum state, then the value of said energy was uncertain. You can’t say how many joules we gained or lost on the “collapse” because we didn’t know how many joules exactly were in $H$. If we measure something that doesn’t commute with $\hat H$, then after the measurement we don’t know how many joules are in $H$. It doesn’t contradict to the conservation of energy. It just means that one cannot apply the law to a state, that isn’t an energy eigenstate, with absolute precision.
Other superpositions are simply lost without trace. They do not carry any energy because in quantum mechanics energy of a state is proportional to the probability of that state. A state that has no probability carries no energy. After the collapse the probability of the other states is zero.