Also according to the different interpretations of quantum mechanics, isn't wave function collapse explained differently in different versions where in some you have branching out into other universes (so paradoxes dont occur.)
Yes, it's an interpretation
. :) That does not make it
true. There are tons of interpretations of quantum mechanics.
The many worlds interpretation you're referencing is also not a method for avoiding causal paradoxes. It simply says that every possible measurement of a system is realized, just in other universes. In the technical jargon, when we "measure a quantum system", its wave function collapses to one of its eigenvalues (of which there might be an infinite number or continuum.) The eigenvalue is the observed quantity -- perhaps the energy, or position, or momentum. Many worlds interpretation says that wave function collapse does not simply choose one of the eigenvalues. It chooses
all of them, each one in a different universe.
Personally, I don't like many worlds interpretation. It doesn't really explain anything. It's just a cute way of thinking about wave function collapse.
Also, in the sense of space or time entanglement, tunneling or teleportation (which has been reported to occur at much faster than c) how can that be explained without breaking down space and time at the very basic levels?
Wave function collapse is instantaneous (to the extent that observations have constrained it), but it does not transfer information! So it does not violate causality. This is highly related to the quantum eraser experiment I linked to.
Another example: it is quite common to encounter waves that have a phase velocity faster than c. But phase velocity of a wave is not what carries information. That's the
group velocity, which is always slower or equal to c. (Actually there are some unusual cases where you can get group velocity faster than c, but only in highly lossy media which make it impossible to use for breaking causality anyway.)
To reiterate, causality is a deep principle that stems from logic more so than any physical law. To have physics without causality is basically to not even have physics.
You can try to think of all sorts of creative ways to violate causality. But if you think about it, you'll find those lead to nonsensical situations, and if you do experiments, you find the universe is incredibly good at preventing it. To such an amazingly thorough degree that it almost becomes freaky.