In your worksheet you are attempting to solve a system of six differential equations by splitting the process into two stages.  In the first stage you solve for the unknowns p, q, r, and then in the second stage you solve for theta, phi and psi which depend on p, q, r.

The preferred way is to solve the entire system of six equations in one pass.  I have modified your worksheet to do that.  Along the way I have made several other minor adjustments.  I believe these should be acceptable to your Maple 13, Let us know if it doesn't work.

Пример02-новый.mw

I make a lot of animations with varying degrees of complexity.  I use Maple to generate frames, and I use ImageMagick to assemble the frames into an animation with complete control on everything.

ImageMagick is an extensive suite of software tools for image manipulation.  It is open and free.  I use it on Linux, but I understand that it also works on other platforms.  In all cases it is accessed through the command-line.

Here I will describe the steps needed to accomplish what you have asked.

1. Save your Maple animation to an (animated) GIF file as you have described.  Let's call that file anim-raw.gif.
2. Apply ImageMagick's convert command to split that file into individual frames:
         convert   anim-raw.gif   frame-%03d.gif
   This will write the individual frames into files named frame-000.gif through frame-nnn.gif.  If you have more that a 1000 frames, change the %03d to %04d.  That will write the files frame-0000.gif, etc.
3. Now put together the individual frames into a single animated GIF called anim.gif:
         convert   -delay 10   -loop 2   frame-*.gif   anim.gif
   The "-loop 2" option says that we want to play the animation twice and then stop.  Change the 2 to anything you wish, or say "-loop 0" if you want the animation to play forever.
   The "-delay 10" option specifies the delay between frames.  The delay is measured in units of 1/100 of a second, so "-delay 10" means 1/10 of a second between frames.

I think that addresses all the questions that you have asked, and as we see, it takes just two commands typed on the command-line.

There are many more options available.  Consult ImageMagick's documentation.

It seems to me that your problem originates in TeXShop.  I don't have a Mac and I have no experience with TeXShop.  I do all my work on Linux, and with TeXLive which comes with it.

I downloaded your Konstruktion_Masterfile.mws, loaded it into Maple 2018, and saved the result as LaTeX through File->Export As...

Then compiled the resulting file through the command-line:

pdflatex Konstruktion_Masterfile.tex      # first pass
pdflatex Konstruktion_Masterfile.tex      # second pass

There were 6 error messages which I ingored.  They all were about missing { and } on line 4779.  I didn't try to fix those; you should be able to do it since you are familiar with the content. 

Here are Konstruktion_Masterfile.tex (which I have renamed Konstruktion_Masterfile.txt) and Konstruktion_Masterfile.pdf.  They look OK to me.

Konstruktion_Masterfile.txt
Konstruktion_Masterfile.pdf

Download mw1.mw

Download mw.mw

Download make_matrix.mw

Before entering the for-loop, define;

mycolors := ["black", "blue", "red", "pink"];

Then specify the color in the for-loop:

for k to 4 do
  R := dsolve(eval({bc, sys}, Ha = L[k]), fcns, type = numeric, AP);
  AP := approxsoln = R;
  p1u[k] := odeplot(R, [y, U(y)], 0 .. 1, numpoints = 100,
      labels = ["y", "U"], style = line, color = mycolors[k]);
end do;

Alternatively, you may omit the color option in the loop, and apply the colors to the composite plot, as in;

display([p1u[1], p1u[2], p1u[3], p1u[4]], color=mycolors);

but note that in the first argument of display I have changed your curly braces to square brackets.

Kitonum has already shown how to compute the derivative for a specific function f. I just want to add that specifying the f is not necessary.  It can be done for any f:

restart;
D[1](f)(z*sqrt(a/nu[f]), U*t/(2*x));

restart;
with(Student[MultivariateCalculus]):
r := (phi,theta) -> <(cos(phi)+3)*cos(theta), (cos(phi)+3)*sin(theta), sin(phi)>;
SurfaceArea(Surface(r(phi,theta), phi=0..2*Pi, theta=0..2*Pi));

The answer is 12 π².

I don't have GlobalOptimization, but that does not seem to be an impediment. The basic minimize() works quite well.

As to plotting, I don't understand the question.  Fun is a function of four variables.  How do you expect to plot a function of four variables?  Furthermore, how is that question related to the minimization issue?

After you have solved the PDE, as in sol := pdsolve(...), do
U := sol:-value():
U(2.0, 3.0);
U(2.0, 3.1);

restart;
M := 5;
F := Matrix(M+1,M+1):
for r from 2 to M+1 do
  for s from 1 to r-1 do
    if type(r+s, odd) then
      F[r,s] := 2^(k+1)*sqrt((2*r-1)*(2*s-1));
    end if
  end do
end do:
evalm(F);

That error message manifests a "misunderstanding" by Maple but it appears to be harmless.  Introduce a third parameter, k, and assign it an arbitrary value and Maple produces the correct result:

restart;
f:=x->piecewise(x>=-1/2 and x<=1/2,1);
eq:={diff(x(t),t)-sum(f(t-k*T),k=0..K)*x(t)=0,x(0)=1};
sol:=dsolve(eq, numeric, parameters=[T,K,k]);
sol(parameters=[1,3,-1]);  # the third argument is an arbitrary number
plots:-odeplot(sol, [t, x(t)], t=0..6,
    color=red, thickness=3, view=0..40);

Or with a more interesting choice of parameters:

sol(parameters=[2,2,-1]);  # the third argument is an arbitrary number
plots:-odeplot(sol, [t, x(t)], t=0..8,
    color=red, thickness=3, view=0..13);

Download mw.mw

Write the summation as

sum( irem(s+r,2) * whatever, s=1 .. r-1);

because irem(s+r,2) is zero when s+r is even, and one otherwise.