I think that all your questions can be answered only numerically with the specified parameter values. Here is an example:

restart;
A:=x->mue+(mun/gama)+(u0^2)-mud*x-mue*exp(x)-(mun/gama)*exp(gama*x)-(u0^2)*(1-2*x/u0^2)^(1/2):

mue:=1/(1+alpha+beta):
mun:=alpha/(1+alpha+beta):
mud:=beta/(1+alpha+beta):
u0:=(mue+mun*gama)^(-1/2):

A1:=eval(A(x),[alpha=0.2,beta=0.3,gama=17]);
r:=solve(op([1,1],indets(A1,sqrt))>=0); # The domain of A1
plot(A1, x=-0.4..op(2,r), size=[800,400]);
fsolve(A1,x=-0.1..0.1); # The first root
fsolve(A1,x=0.1..op(2,r));  # The second root
fsolve(diff(A1,x),x=0.1..op(2,r));  # The root of first derivative

To investigate the dependence of the results on the parameters, see the  Explore command in the help.

restart;
int(tanh(x)/sqrt(x^2+1), x=1..100, numeric);

                                 4.344661055

You have taken too little value for numpoints  (n=100). Try

plot3d( [V,0], 0..2*Pi, 0..2*Pi, numpoints=40000);

Use the inert sum, that is  Sum  instead of  sum :

restart;
X := x->Sum((-1)^m*((1/2)*x)^(2*m+n)/(factorial(m)*factorial(m+n)), m = 0 .. infinity);
diff(X(x), x);  # The derivatives series
applyop((combine@expand),1,%,2*m+n); # Simplification

Edit.

f:=x->sqrt(1+x);
(f@@6)(x);

restart;

Pbc:=proc(n::posint)
local Ind, Var1, Var2;
uses combinat;
Ind:=permute([0$n,1$n],n):
Var1:=[seq(x||i,i=0..n-1)]: Var2:=[seq(y||i,i=0..n-1)]:
{seq(seq(inner(Var1,ind1)*inner(Var2,ind2), ind1=Ind), ind2=Ind)};
end proc:

Examples of use (for n = 2 the result coincides with Carl's one):
Pbc(1);
Pbc(2);
Pbc(3);             

There will be infinitely many such vectors. All of them belong to a plane perpendicular to the original vector. Therefore, the set of all solutions depends on two parameters:

restart;
PV:=(a,b,c)->solve(a*x+b*y+c*z=0,{x,y,z}):

# Examples of use:
V:=<1,2,3>;
PV(V[1],V[2],V[3]);
V1:=eval(<x,y,z>, %); # General solution
V.V1; # Check
V2:=eval(V1,[y=1,z=1]);  # One of the solutions

Define functions as procedures, not expressions:

restart;
h:=x->1/(1+exp(-x));
hh:=(w,x,b)->ln(h(w*x + b));
diff(hh(w,x,b), w);  # Or  D[1](hh)(w,x,b);
normal(expand(%));

I always refer to previous objects using their names or by  %  and so on.

Edit.

Here is a more traditional way:

restart;
IntegrationTools:-Change(int(arccos(x)*arcsin(x),x), x=sin(t));
eval(%, sin(t)=x);

restart;
b:=proc(n)
option remember;
  if n=0 then return 0 elif n=1 then return 1
  else
   (b(n-1)+b(n-2))/2:
  fi;
end proc:

# Examples of use: 

S:=seq(b(n),n=0..10);
plot([$ 0..10],[S]);

You can obtain an explicit formula for the nth member by  rsolve  command:

restart;
# The formula for nth term
rsolve({b(n)=(b(n-1)+b(n-2))/2,b(0)=0,b(1)=1}, b(n));

seq1.mw
 

This is certainly a bug. Do

with(geometry):
point(o, 0, 0);
point(A, 0, 1);
point(d, 0, 2);
point(F, 0.8944271920, 1.4472135960);
line(lOD, [o, d]);
line(lAF, [A, F]);
alpha1 := FindAngle(lOD, lAF);
alpha:=min(alpha1,Pi-alpha1);

                                                alpha := 1.107148718

f:=y->sqrt((1+y)/2):
cos(Pi/2)=0:
cos(Pi/2^n)=(f@@(n-1))(0):

# Examples
cos(Pi/16)=(f@@3)(0);
cos(Pi/64)=(f@@5)(0);

Any command, if you add the  %  symbol in front of it, becomes inert. If we replace  Sum  with  %sum , then the bug disappears. See below

Download simplify_new.mw

Your system contains too many parameters  Q11A, Q11Ay, Q11Az, Q11Iy, Q11Iz, Q11J, Q16A, Q16Ay, Q16Az, Q16Iz, Q16J, Q55A, Q55Ay, Q55Iy, Q66A, Q66Az, Q66Iz  (total 17 ones) with unknown values. If you specify the values of these parameters, Maple easily solves this system. I took parameter values from 1 to 17.

Download copy1_new.mw

Here is a procedure for this:

LeastDegree:=proc(P::polynom, var::{set,list})
local L;
L:=sort(map(p->degree(p,var),[op(P)]));
select(p->degree(p,var)=L[1],P);
end proc:

Examples of use:

f:= 100*x^2*y^2 + 35*x^2*y + 45*y:
g:= 13*x^2*y^2 + x*a*y^2 + 2*y*x^2:
LeastDegree(f, [x,y]);
LeastDegree(g, [x,y]);
                                            45*y
                                     a*x*y^2+2*x^2*y

Edit.