You can use an inert form for subtraction to keep the expression unchanged. For any calculations and transformations use the  value  command:

restart;
expr:=arccos(p%-a);
value(expr);
value(eval(expr,[p=0,a=1]));

_B1  is a binary variable that is 0 or 1 :

restart;
ode:=diff(y(x),x)-y(x)/x+csc(y(x)/x)=0;
sol:=dsolve([ode,y(1)=0]);
about(_B1);
sol1:=eval(sol,_B1=0);
sol2:=eval(sol,_B1=1);
odetest(sol1,ode);
odetest(sol2,ode);

Here is a solution in Maple:
 

Download nlsystem.mw

restart;
plots:-animate(plot,[cos(2*theta),theta=0..a, coords=polar], a=0..2*Pi);

We can significantly increase efficiency if we use symmetries: from one solution we can get  6 * 6 = 36 solutions by permutations of the first three and last three numbers. Instead of loops, it’s slightly more efficient and more compact to use nested seq . So we get the unique solution:

Download isys.mw

Your surface is a torus:

restart;
plot3d(eval([x,sqrt(x^2+y^2)*cos(s),sqrt(x^2+y^2)*sin(s)],[x=cos(t),y=2+sin(t)]), t=0..2*Pi, s=0..2*Pi, scaling=constrained, labels=[x,y,z]);

Download shade.mw

Here is another shorter way:
 

Download shade1.mw

Here is a more automatic plotting for the intersection:

restart;
r1:=2:  r2:=2*(1-cos(theta)):
plot(min(r1,r2), theta=0..2*Pi, color=green, coords=polar, filled, scaling=constrained);

Edit.

The following approach is sometimes useful:

restart;
eq := x^2+floor(x)-10:
Student:-Calculus1:-Roots(eq);
identify(%);

The animation you see here is saved in GIF format. Each polygon from the list remains on the display for exactly 1 second (10 frames per second by default):

restart;
OneFrame:=proc(n)
local Circle, Polygon, Text, S;
uses plottools, plots;
Circle:=plot([cos(t),sin(t),t=0..2*Pi], color=grey, thickness=2);
Polygon:=polygon([seq([cos(2*Pi*k/n),sin(2*Pi*k/n)], k=1..n)], color=yellow);
S:=evalf(n*sin(2*Pi/n)/2);
Text:=textplot([[-0.7,-1.1,"Area of Circle = 3.14"],[0,-1.1,typeset("n","=",n)],[0.7,-1.1,typeset("Area of Polygon","=",evalf[3](S))]], font=[times,16]);

display(Text,Circle,Polygon, scaling=constrained, axes=none, title="Archimedes Approximation for Pi \n", titlefont=[times,bold,18]);
end proc:

plots:-display(seq(OneFrame(n)$10, n=[3,6,9,12,15,18,21,24,27,30,60]), insequence, size=[600,600]); 

Download diffeq.mw

Maple has the command  Student:-LinearAlgebra:-LinearSolveTutor  that solves systems of linear equations step by step, but unfortunately only if the matrix of the system is no more than 5 by 5. Below is a step-by-step solution using the Jordan Gauss method using my program  JordanGausse (all comments are in Russian):

system.mw

reduce the range along the vertical axis (y-range) while maintaining  x-range. Also use a list rather than a set when specifying functions so that all options match the corresponding function:

restart;
plot([0, 2*x^2, 2*x^2 - 2*x^3 + 8/3*x^4 - 4*x^5], x = -10 .. 10, y=-1000..1000, color = ["DarkGreen", "CornflowerBlue", "Burgundy"], axes=box);

Here is another way by the using the  applyrule  command:

restart;
expr:=1/exp(z)*arcsinh(x*exp(C[1]))+x*sin(exp(x))+3*exp(C[1]*y)*sqrt(sin(exp(3*h)));
applyrule(exp(t::anything)=Z, expr);

I think that there are infinitely many ellipses inscribed in a given triangle. But among them there will be only one that touches the sides of the triangle in their midpoints. It is called the Steiner ellipse. See  https://en.wikipedia.org/wiki/Steiner_inellipse

Here is an example of plotting this ellipse:
 

As for orthoptic circle for an ellipse see  https://en.wikipedia.org/wiki/Orthoptic_(geometry)#:~:text=Generalizations%3A,fixed%20angle%20(see%20below).&text=Thales'%20theorem%20on%20a%20chord,two%20points%20P%20and%20Q.

Download Steiner_ellipse.mw

For a rational function  f(x)=P(x)/Q(x)  to have a horizontal asymptote  y=y0  and  y0<>0 , it is necessary and sufficient that the polynomials  P(x)  and  Q(x)  have the same degree. In this case, the asymptote will be the same at + infinity and -infinity. Thus, a rational function cannot have two different horizontal asymptotes. Therefore, your conditions may be only partially implemented.

Here is an example:

Download rational_function.mw