Q:

How could a very strong impact on a planet create deformations on the opposite side?

Update (6/20/19):
I have received some great answer here and they all mostly point to the fact that while effects on the opposite side of the planet are possible, the deformation may or may not resemble a crater but more likely an earthquake or fault zone. I have picked the answer that I think best describes this process in a way that I can build upon for use in my world.
However, my question does ask how it could happen not if it was possible (perhaps this was not clear). In my second set of bullet points I tried to express that I am open to a non-Earth like planet - or even something unlike any planet we know of so long as it could form in theory according to physical laws. I am not sure if I will get an answer satisfying this criteria but this would be the ideal answer. 

In a previous question, I asked about mountains, rivers, trees, and so on forming within an impact crater surrounded by desert. I left it open about how this crater was formed. 
Note: A question similar to mine has been asked here, but the answers do not help me.
Let's imagine that a large, circular or spherical object impacts a planet on one side. The desert mentioned in the previous question exists on the opposite side of the planet from the site of impact. Are there conceivable conditions under which:

The impact would travel through the planet and cause a crater on the other side. The crater does not need to be exactly the same shape as the impact made on the surface of the planet by the object but should be similar.
The impact would emanate outward in a cone-like fashion such that the diameter of the crater on the other side is greater than the diameter of the circle object that made the impact.
The heat (or pressure) is intense enough for at least some of sand near the rim of the crater on the other side to turn into glass.
Mountains of some kind could form near the edge of the crater due to the impact. If this is not possible, then hopefully there is some sort of mantle and crust that allows the gradual formation of mountains (I don't know if this is exactly how that works).

In order to achieve these specifications, let's say that:

The impacting object may be made of any type of material, have any mass, and can impact at any velocity (though I'd like some kind of lower bound).
The planet's internal layers can be made of any type of matter in any phase so long as its still internally consistent with the basic laws of physics and what is known or speculated about planet formation and stability.

I would like a hard-science answer if possible. However, since this is for a fantasy (with a bit of sci-fi) setting, I am okay with some hand-waving if it comes to that. This event would have happened in the far history of the world so its precise nature would not necessarily be known or understood.

A:

Antipode damage?  Yes.  Antipode crater?  No.

Seismic  shaking.  the  weakening  shock  wave  eventually  degrades  into  elastic  waves. these  elastic  waves  are  similar  in  many  respects  to  the  seismic  waves  produced  by  an earthquake,  although  impact-generated  waves  contain  less  of  the  destructive  shear-wave energy than earthquake waves. the seismic waves produced by a large impact may have signiicant  effects  on  the  target  planet,  creating  jumbled  terrains  at  the  antipode  of  the  impact site if they are focused by internal planetary structures, such as a low-velocity core.
this  effect  has  been  observed  opposite  caloris  Basinon  Mercury  and  opposite  Imbrium and orientale on the Moon. the equivalent richter magnitude M caused by an impact of energy E (=1/2 mpvi2) is given approximately by:M= 0.67 log10E – 5.87.

Which is a fancy way of explaining that the force of the initial impact gets all jumbled up on its way through the planet such that what comes out at the antipode isn't circular, but resembles an earthquake.
The source of that quote is from an unknown to me book by H. J. Melosh, Chapter 6, that appears to have been made available for a geology class at the University of Chicago.  If you look at the /docs directory, you'll find a plethroa of information.
Conclusion
You can get antipode damage, but you cannot get the formations you're looking for.  Even dealing with a sphere like the moon, which can be somewhat thought of as homogenously solid (rather than having solid and liquid layers), you don't get the kind of damage you're looking for.
Keep in mind that this needn't stop you — unless for some reason your story needs to pass muster in front of a bunch of geology PhDs.

A:

Yes, it can happen. When the shock-wave of the impact propagates into the planet, it sooner or later reach the other end. Due to the diffraction of the waves into the planet, it will happen that the wave is focused more in certain areas, dealing major damages.
However, it would not create a crater, but rather an increased fracture of the crust and possibly a bulge. Only if there is some underground collapse, you might get a small crater.
You can find more in this answer

A:

There is geological evidence on Earth that impacts have caused fault zones and earthquakes, leaving permanent features on the other side of the planet.
The mechanism is focusing of the seismic waves -- one kind of wave (p waves, as I recall, the ones that are compression like sound in air) travels completely through the Earth, but it travels at different speeds in different densities of material -- and reflects off discontinuities, just as light or sound will do from a sharp boundary with different speeds on the two sides.
The end result is that, although the refraction of these waves curves them upward, they then reflect downward from either the crust-mantle boundary or the mantle-outer core boundary.  Overall, this has the effect of focusing these waves into a fairly small area (much smaller than a continent) on the opposite side of the Earth from the impact.  This area will then experience seismic events a sizable fraction of the magnitude of the direct impact events -- but of course without the vapor/ash cloud, ejecta, heat blast, and surface waves.
I've read a hypothesis that the Permian Traps, blamed for one of the great extinction events, were created by an impact larger than the Chicxulub event in the south Pacific (where we have little chance of seeing the impact footprint directly because the sea floor there has subducted since).

