Say, if I detonate a ridiculously large nuke on this side of the Earth, is it possible that the shock wave travels through the center of the Earth and create an earthquake on the other side? If yes, how much of force are we looking at?
The issue is not the strength of the impact, but the structure of Earth.
That means that an impulse strong enough to pass the mantle and outer core would get diffused, because the pressure would be applied in all directions. Of course, an impulse strong enough would affect the terrestrial crust and mark it, but that effect would not necessarily appear in the opposite point of impact; most likely the energy would be dispersed breaking the weaker parts of the crust.
Yes and no. Yes that in nuclear explosions, like earthquakes, are routinely detected by seismographs. (Search on "seismic detection of nuclear explosion" for more info.) No because, as has been pointed out, the Earth has a liquid outer core, and most seismic waves won't propagate through that. They'll go around, travelling through the solid part of the mantle.
(This is pretty simplistic: if you want more, look for a basic text/web page on seismology.)