Is there a connection between the spreading of the wave packet of the universe and the time arrow? Entropy of position and wave packet spreading.
Why do wave packets spread out over time?
Wave packet of the universe and its  spreading.
Is there a connection between the spreading of the wave packet of the universe and the time arrow (second law of thermodynamics), despite the reversible character of physical laws?
 A: No. The arrow of time implied by taking entropy seriously is due to the molecular chaos asssumption normally made when deriving the Boltzmann equation. Poincare proved mathematically that a mechanical system will return arbitrarily close to its inital state (Poincare recurrence). But, the time scales for this to occur are unthinkably large compared to the current age of the universe. 
A: I think it will depend on the model of the universe. The arrow of time is classically defined in thermodynamics as the other answer states. At the classical particle level it is connected with the number of microstates in a macrostate , and also quantum statistical mechanics allows to extend the definition to quantized states, thus extends the thermodynamic definition to a quantized universe.
Any model of  the universe at the early times where it might be considered a "wave packet" will have a number of microstates in the macrostate called "wave packet" in your question. In this sense in the Big Bang model there will be created continuously new microstates by the expansion, and thus the answer is positive, since the microstate number increases with the expansion, thus entropy will be increasing.
