What is the minimum phase shift for Quadrature Amplitude Modulation (QAM)? I'm a bit of a noob here, looking for a simple formula or just a table which can tell me the absolute minimum phase shift angle possible for different variations of QAM. Particularly QAM 16, 64, 256 and 1024. If the answer is zero (is it?), then what is the minimum phase shift possible in each case other than 0?

Additionally, what is the limiting factor which governs the rate at QAM can switch between different phases? In other words, what is the maximum rate at which QAM can switch between different phases, and how is that determined for a particular device? Is there a relation to frequency of transmission (say a 1 GHz signal), or are frequency of transmission and phase-switching rate completely independent?
The minimum phase shift in 16-QAM, 64-QAM, 256-QAM etc is 0 (same value regardless of whether you are using degrees or radians!) and occurs when the symbol transition is from one constellation point on some radius vector to another constellation point on the same radius vector.  If we take the constellation points as having coordinates $(i,j)$ where $(i,j) \in \{-3, -1, +1, +3\}$ in 16-QAM, then $(1,1)$ and $(3,3)$ lie on the same radius vector (line of slope 1 through the origin) and so the RF phase remains unchanged in the transition from one constellation point to the other. Obviously, the same example will also work for 64-QAM, 256-QAM, and 1024-QAM.

What is the minimum non-zero phase shift?

Well, for 16-QAM, the constellation points $(1,3)$ and $(3,3)$ are distance 2 apart, and together with the origin, form an obtuse triangle with sides of lengths $2$, $\sqrt{10}$, and $\sqrt{18}$. From the standard formula
$$c^2 = a^2 + b^2 - 2ab\cos C$$ relating the sides $a$, $b$, $c$ of a triangle opposite to angles $A$, $B$ and $C$, we get that
$$2^2 = 10 + 18 - 2\sqrt{180} \cos C$$
giving $\arccos \frac{2}{\sqrt{5}}$ as the minimum nonzero phase shift in 16-QAM.
Similar calculations for 64-QAM, 256-QAM and 1024-QAM as left as an exercise for the reader.