How to speed up Manipulate with Graphics3D

I played with this a bit to try to speed it up. Usually I don't like touching code that is this long, but your code was well organized so it was easy to see what was going on, and it was easy to remove parts and test them separately.

What I did first was to remove all optimization attempts that I was not convinced had any significant effect. Let's find out what part of the code is causing the slowdown before trying to optimize it.

So, I removed memoization, TrackedSymbols (seems unnecessary), Dynamic (unnecessary).

Then I changed ψ and θ to have continuous values instead of discrete. Discrete values were not needed since we removed memoization, and having discrete angles creates makes the animation look choppy, even if performance is good. So now I had {{ψ, 60}, 0.1, 90.}, {{θ, 40}, 0.1, 90.} in Manipulate. Also note that using integers in computations is slower than using machine precision numbers. Using discrete values had the side effect of doing all computations with integers.

Next, I changed all Sequence to List, the reasoning being that Sequence seemed unnecessary and would cause some additional computations to be done (though I really doubt those had an effect on performance).

Next, I removed graphicsBox and related things. To keep the plot area constant, I added the following options to Graphics3D:

 PlotRange-> 2 {{-1,1},{-1,1},{-1,1}}, SphericalRegion -> True 

Side note: This is unrelated to performance or your question, but I would prefer the ViewPoint to be closer to the origin. The large distance causes the perspective to be so flat that I find it hard to mentally follow the rotation. This is of course your decision :-)

Next, I removed the separate objects in the Graphics3D to see which had the greatest performance effect. planeSurfaceFirstRotation and planeSurfaceSecondRotation seemed to have a stronger effect on performance than the other two, so I decided to optimize surfaceX2:

surfaceX2[α_] := With[{r=1.5}, {EdgeForm[None], Polygon@Join[ Table[{r Cos[u],r Sin[u],0}, {u,0,α,0.1}], {{r Cos[α],r Sin[α],0}, {0,0,0}}]} ]; 

Now it returns graphics primitives, so I removed the [[1]] part from functions that use it. Another note here: you had Axes -> None, Boxed -> True, Lighting -> {{"Ambient", Blue}} in your parametric plot command. These had no effect at all because the graphics primitives were extracted from the result and placed in a different Graphics3D.

At this point dragging the Manipulate sliders updates the graphics with the same frame rate as rotating it with the mouse. So I assume that at this point the performance is limited by rendering the graphics on screen, not by building the symbolic Graphics3D object. The easiest way to speed things up is to avoid transparent objects. If I make all objects opaque, the animation is completely smooth on my computer. Personally, I prefer the looks with everything opaque except the surfaceX2, but removing all opacity (including from the box) will speed things up noticeably.

Here's your code, modified as I described, except for removing opacity (some cleanup will be needed as I didn't format it as well as you did). On my machine the end result is noticeably (but not very significantly) smoother than with the original code.

Manipulate[ Graphics3D[{ (* Original coordinate axes *) origCoordAxes, (* Vectors rotated once *) vectorsRotatedOnce[ψ], (* Vectors rotated twice *) vectorsRotatedTwice[ψ, θ], (* Plane surface first rotation *) planeSurfaceFirstRotation[ψ], (* Plane surface second rotation *) planeSurfaceSecondRotation[ψ, θ] }, BoxStyle -> GrayLevel[.75], ViewVertical -> {0, 0, -1}, ViewPoint -> {-12, -10, -5}, PlotRange-> 2{{-1,1},{-1,1},{-1,1}}, SphericalRegion->True ], {{ψ, 60}, 0.1, 90.}, {{θ, 40}, 0.1, 90.}, Initialization :> ( x0 = {2, 0, 0}; y0 = {0, 2, 0}; z0 = {0, 0, 2}; x1[ψ_] := RotationMatrix[ψ Degree, z0]. x0; y1[ψ_] := RotationMatrix[ψ Degree, z0]. y0; z1[ψ_] := RotationMatrix[ψ Degree, z0]. z0; x2[ψ_, θ_] := RotationMatrix[θ Degree, y1[ψ]]. x1[ψ]; y2[ψ_, θ_] := RotationMatrix[θ Degree, y1[ψ]]. y1[ψ]; z2[ψ_, θ_] := RotationMatrix[θ Degree, y1[ψ]]. z1[ψ]; origCoordAxes = List[ Black, Opacity[.25], Arrow[Tube[{{0, 0, 0}, {1, 0, 0}}, 0.005]], Text[Style["\!\(\*SubscriptBox[\(x\), \(0\)]\)", Black, Large], {1.2, 0, 0}], Arrow[Tube[{{0, 0, 0}, {0, 1, 0}}, 0.005]], Text[Style["\!\(\*SubscriptBox[\(y\), \(0\)]\)", Black, Large], {0, 1.2, 0}], Arrow[Tube[{{0, 0, 0}, {0, 0, 1}}, 0.005]], Text[Style["\!\(\*SubscriptBox[\(z\), \(0\)]\)", Black, Large], {0, 0, 1.2}] ]; vectorsRotatedOnce[ψ_] := List[ Red, Opacity[1], Arrow[Tube[{{0, 0, 0}, x1[ψ]}, 0.01]], Text[Style["\!\(\*SubscriptBox[\(x\), \(1\)]\)", Black, Large], (1.2 x1[ψ])], Text[Style["ψ", Lighter@Blue, Large], ((1/3) ( x1[ψ] + x0))], Green, Arrow[Tube[{{0, 0, 0}, y1[ψ]}, 0.01]], Text[Style["\!\(\*SubscriptBox[\(y\), \(1\)]\)", Black, Large], (1.2 y1[ψ])], Text[Style["ψ", Lighter@Blue, Large], ((1/3) ( y1[ψ] + y0))], Blue, Arrow[Tube[{{0, 0, 0}, z1[ψ]}, 0.01]], Text[ Style["\!\(\*SubscriptBox[\(z\), \(1\)]\)", Black,Large], (1.2 z1[ψ])] ]; vectorsRotatedTwice[ψ_, θ_] := List[ Lighter@Red, Opacity[.3], Arrow[Tube[{{0, 0, 0}, x2[ψ, θ]}, 0.01]], Text[ Style["\!\(\*SubscriptBox[\(x\), \(2\)]\)", Black, Large], (1.2 x2[ψ, θ])], Text[ Style["θ", Lighter@Green, Large, Opacity[1]], ((1/3) ( x2[ψ, θ] + x1[ψ]))], Lighter@Green, Arrow[Tube[{{0, 0, 0}, y2[ψ, θ]}, 0.01]], Lighter@Blue, Arrow[Tube[{{0, 0, 0}, z2[ψ, θ]}, 0.01]], Text[Style["\!\(\*SubscriptBox[\(z\), \(2\)]\)", Black,Large], (1.2 z2[ψ, θ])], Text[Style["θ", Lighter@Green, Large, Opacity[1]], ((1/3) ( z2[ψ, θ] + z1[ψ]))] ]; planeSurfaceFirstRotation[ψ_] := List[ Opacity[.2], Directive[Blue, Glow[Blue], Specularity[0]], Rotate[surfaceX2[ψ Degree], 0, {0, 0, 1}], Rotate[surfaceX2[ψ Degree], π/2, {0, 0, 1}] ]; planeSurfaceSecondRotation[ψ_, θ_] := List[ Opacity[.2], Directive[Green, Glow[Green], Specularity[0]], Rotate[Rotate[surfaceX2[θ Degree], ψ Degree, {0, 0, 1}], 3 π/2, x1[ψ]], Rotate[Rotate[surfaceX2[θ Degree], ψ Degree - π/2,{0, 0, 1}], 3 π/2, x1[ψ]] ]; surfaceX2[α_] := With[{r=1.5}, {EdgeForm[None],Polygon@Join[Table[{r Cos[u],r Sin[u],0},{u,0,α,0.1}],{{r Cos[α],r Sin[α],0},{0,0,0}}]} ]; ) ]