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

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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.

Asking for a phase portrait for your equation is not exactly meaningful.  The responses that you have received present graphs of solutions y(x) versus x, which is not the usual meaning of a phase portrait.  A phase portrait is a plot in the phase space.

The concept of the phase space, however, pertains to autonomous differential equations.  Your equation is non-autonomous, so the graphs that you have been presented with are remotely reminiscent of phase portraits but you should be aware that they are not phase portraits.

That said, here is what I would do if I were to plot a set of graphs of y(x) versus x (not a phase portrait).

>  restart; >  with(DEtools): >  de := diff(y(x),x)=x^2-y(x)^2; >  a := 2.0;  b := 3.0; >  ic := seq(y(s)=+a, s=-a..a, 0.3),       seq(y(s)=-a, s=-a..a, 0.3),       seq(y(-a)=s, s=-a..a, 0.3),       seq(y(+a)=s, s=-a..a, 0.3): >  DEplot(de, y(x), x=-b..b, [ic], y=-b..b,     linecolor=black, thickness=1, arrows=none);

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