If a very huge Earthquake occured anywhere on Earth could waves emerge to come together again on the opposite side?
It is called "antipodal focusing". See for example Antipodal focusing of seismic waves observed with the USArray.


  We present an analysis of the M-w = 5.3 earthquake that occurred in the Southeast Indian Ridge on 2010 February 11 using USArray data. The epicentre of this event is antipodal to the USArray, providing us with an opportunity to observe in details the antipodal focusing of seismic waves in space and time. We compare the observed signals with synthetic seismograms computed for a spherically symmetric earth model


The above paper deals with "body waves" that travel through the interior of the Earth.

There are also Rayleigh waves that travel on the surface and can travel around the Earth several times before dissipating (Wikipedia). Antipodal focusing of seismic waves due to large meteorite impacts on Earth does numerical simulations of surface waves at the antipode of the Chicxulub impact. The waves do not arrive at the antipode at the same time because of Earth’s ellipsoidal shape and different rock properties along their paths.


  
  Isosurfaces of the norm of the peak displacement vector after a vertical impact for the impact hemisphere (left-hand side) and antipodal hemisphere (right-hand side).  


(They say: "Note that contours of the continents are today’s, not their end Cretaceous positions, and therefore do not show the original Chicxulub antipode 65 Ma.")