Proof that a spherical lens is stigmatic In geometric optics, we generally allow that, for example in the case of a convex lens, rays coming from a particular point get refracted towards another particular point on the opposite side of the lens.
How is this proven from Snell-Descartes' law? Do we need to use the paraxial approximation in order for this to be true, or is it actually exactly true?
 A: Usual spherical lenses are only approximately stigmatics, that is to say the image of a source-point is itself a point. You need paraxial approximation to ensure approximate stigmatism and avoid spherical aberration.
You can see it by playing around with just a plane interface between two different transparent media like the surface of water. Using paraxial approximation, you can prove the relation $$\frac{n}{HA} = \frac{n'}{HA'}$$ where $n$ is the refraction index of the medium where the object $A$ lies (for example a fish, then $n=1.33$), $H$ is the orthogonal projection of $A$ on the interface, $A'$ is the image and $n'$ is the refraction index of the medium where the observer thinks $A'$ is (for example the air so $n'=1$). If the fish is $40$ cm under the water, you will see it's image only $30$ cm away from the interface (that's why it's difficult to catch a fish in an aquarium on first try).
As you need spherical interfaces to build your lens and it's not even working with planar interfaces (which are a particular case of spherical interfaces), it can not be true for any type of spherical lens.
Here are two animations you can try to see how stigmatism works:


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*one with plane interface

*and another with spherical mirror
A: According to an article by the Optometric Science Research Group titled Stigmatic optical systems:

"There would appear to be little disagreement on what constitutes an astigmatic system in the case of a thin lens: the cylinder is not zero. A spherical thin lens is stigmatic or not astigmatic. The issue is less clear in the case of a thick system. For example, is an eye stigmatic merely because its refraction is stigmatic (spherical)?"

According to Snell's law, (from Wikipedia) the largest possible angle of incidence which results in a refracted ray is called the critical angle. When something exceeds this angle no there is no refraction point for the object in the image, meaning the result is astigmatic.
Also according to Stigmatic optical systems:

"An eye may be astigmatic despite having a stigmatic refraction."

A: Actually there is one imaging mode where the spherical lens is a perfect lens: converging all rays, no matter how far from paraxial, to a single focal point. This is the case of the aplanatic sphere, and it is extremely important in the design of microscope objectives.
We're talking about homogeneous and isotropic spheres here. There are other perfect spherical lenses which have nonuniform refractive index. This is the class of Luneburg lenses, of which the Maxwell Fish Eye is an example. This class of lenses make the points of certain concentric spherical surfaces perfectly conjugate, that is, rays from a point on one surface will converge precisely to a point on the other.
Back to the aplanatic sphere, sketched below: 

Here we have a sphere of refractive index $n_2$ steeped in a medium of index $n_1$.  We consider any point $P$ on the sphere concentric with the lens but with a radius $n_1/n_2$ times that of the spherical lens. This point then has a perfect virtual image at $Q$, with no aberration whatsoever, i.e. all the rays output to the left of the device converge precisely to the point $Q$ if extended. 
This principle is applied in high numerical aperture, dipping or oil immersion microscope objectives, as sketched in the figure below taken from Born and Wolf, "Principles of Optics". 

The object surface is the sphere inside the index matching fluid with the same relationship to the ball lens at the microscope end as we have discussed above. The point $P$ on this sphere has a perfect virtual image at $P_1$. Moreover, the numerical aperture of the virtual image is reduced by a facgtor of $n_2^2/n_1^2$. The meniscus lens's inner surface is concentric with the image sphere through $P_1$; accordingly, rays from any point source on the first imaging surface are undeviated by the meniscus's inner surface. So now we can play the aplanatic sphere trick again with the meniscus's outer surface, and we get a perfect image of $P$ at $P_2$. 
Repetition of this trick, each time lowering the numerical aperture by a factor of $n_{i+1}^2/n_i^2$ swiftly brings the field from any point on the object surface close to collimation, and with zero aberration. It is now a simple matter to make the lens collimating with extremely low aberration for any point on the spherical object surface within the index matching medium. 
