Does universal wave function exist? Does universal wave function exist? 
What the science tells us?
 A: Wavefunctions live on configuration spaces.  A configuration space is (not surprisingly) the space of configurations of the system in question.  In elementary quantum mechanics with several particles, it's the space of configurations of those particles.
In a more field theoretic scenario, the functional Schroedinger picture, you have wavefunctions (strictly wavefunction als) on the configuration space of classical fields.
So the meaning of a wavefunction depends on the elements in your model (particles in QM, classical fields in field theory).  I guess by "universal wavefunction" you mean a wavefunction which describes everything in the universe?  If the principles of quantum mechanics are correct and universal, then once you have identified the ingredients of a theory of everything, then there will be a wavefunction to describe its quantum mechanical states.
A note of caution:  when you hear people (e.g. Hartle and Hawking) refer to "the wave function of the universe", they're not referring to this ultimate entity, but rather to a wavefunction which describes the large scale structure of a highly symmetric model of the universe.  Their configuration space is minisuperspace, and their wavefunction usually refers to just the allowed values of a single parameter!
