I haven't examined / debugged your code since it seems to me much too complicated for what it's supposd to do. Instead, I wrote the following alternative which does what you want. This is a slightly edited version of what I had initially posted.

Download piston-crank-animation2.mw

The format of the font specification is defined in ?plot,options under the "font" entry. Your specification axesfont=[12,12] does not fit that specification. Maple does not complain about that in the worksheet mode. I would call that a shortcoming of the worksheet-mode parser.  The commandline-mode parser correctly flags it as invalid.

I suppose that by axesfont=[12,12]  you wish to select 12-point fonts for the horizontal and vertical axes.  To see that Maple's worksheet-mode parser interprets that differently, change that to axesfont=[36,8].  You will see that the "36" is ignored.

The try/catch mechanism may be what you are looking for.

Download mw.mw

Your code expresses the essence of the solution, but it can be streamlined a bit.

restart;
eq := <x, y> =~ lambda*<1, 1> + mu*<1, 2>;
convert(eq, set);
solve(%, {lambda, mu});

We see that solve() returns a unique solution, indicating that any vector <x,y> may be uniquely represented as a linear combination of <1,1> and <1,2>, indicating that those form a basis for the space.

Maple's PDF export has been broken in the last several releases. Despite repeated complaints on this forum, no solution seems to be forthcoming.

These two workarounds work quite well on the command-line in Linux.  The equivalent command lines may be available in Mac and Windows but I have neither so I wouldn't know.

1. Workaround #1:  Save the graphics as EPS, let's say filename.eps, then do
       epstopdf filename.eps
   to produce filename.pdf.
2. Workaround #2:  Save the graphics as SVG, let's say filename.svg, then do
       inkscape --export-type=pdf --export-area-drawing filename.svg
   to produce filename.pdf.

Both methods produce good quality PDF.  The file size of the PDF file in method #2 is about 1/20 of that of method #1 but the latter is more faithful to the original.

I have attached PDFs cooresponding to 
    plot3d(x^2+y^2, x=-1..1, y=-1..1);
converted through the two methods.

parabola-from-eps.pdf
parabola-from-svg.pdf

Here is the solution, but don't just pass it on to your teacher. She won't believe you.

To get an answer with detailed explanation, show what you know and how far you have gotten on your own on this homework.

restart;
f := x -> (x^3 + 9*x^2 - 9*x - 1) / (x^4 + 1);
numer(D(f)(x) - 1/2);
xvals := fsolve(%, x, maxsols=4);
seq(f(t) + 0.5*(x-t), t in xvals);
plot([f(x), %], x=-7..7, scaling=constrained, color=[seq(cat("oldplots ", i), i=1..5)]);

restart;
with(plots):
with(plottools):
p1 := plot(sin(x), x=-2*Pi..2*Pi, color="Red"):
display(
	plot(sin(x), x=-2*Pi..2*Pi, color="Green"),
	line([-2*Pi,0], [2*Pi,0]),
	line([0,-2*Pi], [0,2*Pi])
):
p2 := rotate(%, Pi/4):
display(p1,p2, scaling=constrained);

I don't see a simple way of adding tickmarks.

restart;
with(plots):
vertices := [
	c__1 = <0, -2>,
	c__2 = <1, 2>,
	c__3 = <2, 2>,
	c__4 = <0.5, 6>,
	c__5 = <1, 4> 
]:
Labels := map( v -> [convert(rhs(v), list)[], lhs(v)], vertices):
display(
	textplot(Labels, align={above}),
	plottools:-polygon(convert~(rhs~(vertices), list), style=line, color=red)
);

See https://www.mapleprimes.com/questions/229945-Simulation-Of-A-Solution-Of--Neutral for a lot of ideas on how to handle delay differential equations.

I don't see what you mean by tracing the moving points since the number of point changes in your sequence of plots. Here is how to draw traces of individual moving points that preserve their identities during the motion.

restart;
with(plots):
f := t -> sin(t);
g := t -> [cos(t), sin(t)];
trace_me := proc(t)
	display(
		pointplot([t,f(t)], symbol=solidcircle, symbolsize=30, color="Red"),
		plot(f(s), s=0..t, color="Green"),
		pointplot(g(t), symbol=solidcircle, symbolsize=30, color="Orange"),
		plot([g(s)[], s=0..t], color="Blue"),
	scaling=constrained);
end proc:
animate(trace_me, [t], t=0..2*Pi);

I will illustrate the required procedure in terms of a function that I made up. If this does not work in your case, then you need to provide more details about your function f_p(x).

Download area-under-envelope.mw

You are interpreting Maple's result as if it had said y(x) = whatever.  But that's not what the answer says.  Rather, the answer is an integral with the upper limit as y(x).  The following worksheet shows how to verify that Maple's solution does indeed satisfy the differential equation.

Download mw.mw

If we plot the left-hand side and right-hand side expressions of your equation, we see that the plots don't intersect, and therefore there is no (real) solution to your equation.

eq := exp(x) * (1 + sqrt(1 + exp(-2*x))) - arcsinh(exp(-x)) = x;
plot([lhs(eq), rhs(eq)], x=-3..3);

Here is how to go about it.  The resulting figure is not exactly like what you have shown which indicates that your figure was produced based on a different set of parameters than what you have provided.

Download mw.mw