Why is the L3 Lagrangian point not perfectly stable?
In the circular restricted three-body problem (CR3BP or CRTBP) an object at any of the first Lagrange points L1, L2, L3 is unstable mathematically. Yes, a ball at the exact top of a hill will sit there, but any tiny offset in position or tiny nonzero velocity will lead to it accelerating down the hill.
As @JamesK points out these three points are unstable against displacements along the Sun-Earth line, but stable against displacements in the perpendicular direction.
That's why the diagram shows blue arrows along the Earth-Sun line and red arrows perpendicularly for these three points.
It is important to point out that this is the zero-velocity (pseudo)potential surface. The diagram for stability against small velocities will look different!
And why is the Earth-Sun L3 point a bit less than one A.U.?
The Lagrange points are mathematical points in the CR3BP where forces exactly cancel. They can be calculated in either an inertial frame (where the Earth and Sun rotate around their common center of mass) or in a rotating frame where centripetal and gravitational forces exactly cancel.
The equations for this are a little complicated when you try to solve for the exact positions. They are given for example in Wikipedia's Lagrange point; mathematical details. It is important to remember that the two massive bodies orbit around a common center of mass rather than the smaller one orbiting exactly around an immobile larger one.
The big $R$ is the distance between the two massive bodies, and the little $r$ is the distance from a given Lagrange point to the smaller one which is in this case, from Earth.
The locations are where the forces cancel, and for L3 that cancellation happens at...
uhoh! The equations in that Wikipedia link for L3 seem wrong!
Looking elsewhere in the internet I've found conflicting solutions for the location of L3. The solution requires finding roots of a fifth-order polynomial which is hard on paper but may be easy with a computer (Wolfram alpha, Maple, Mathematica...) or just by numerical searching.
But considering the disagreements I've found I'm not sure right now what the correct polynomial even is, so I'll have to derive it.
stay tuned for updates!