There is no list indexing needed at all:

restart;
X:= eval(
    <
        1,             -X3_3/2 - 1/2, 0,   -X2_3;
        -X3_3/2 - 1/2, -2*X3_4 - 1,   X2_3, 0   ;
        0,              X2_3,         X3_3, X3_4;
        -X2_3,          0,            X3_4,    1
    >,
    [X3_4, X3_3, X2_3]=~ [1-A^2, -1-A^2, A*(3*A^2 - 1)] /~ (3*A^2 - 1)
);     
andseq(
    andseq(
        is(e::nonnegative),
        e= LinearAlgebra:-Eigenvalues(eval(X, A= r))
    ), 
    r= solve(A^3 - A)
);

I wasn't sure about your question #3. The above returns true iff all 12 eigenvalues are real and nonnegative. Any variation of that that you want can also be done without indexing.

Note that is(e::nonnegative) is sufficient to check whether e is real and nonnegative.

Change dis play to display.

Here is a simple solution using select:

M3[select(M3[.., 3]=~ 2, [$op([1,1], M3)])];

To select columns whose 2nd row is 80 (for example), do

M3[.., select(M3[2]=~ 80, [$op([1,2], M3)])];

In either case, the = can be replaced by any of <, <=, >, >=, in, or <>.

For 2D Input, you need to put 1.. after $, as in 

M3[.., select(M3[2]=~ 80, [$1..op([1,2], M3)])];

Appending assuming complex is sufficient to remove the guaranteed nonzero factors in this case. A simple procedure for it is

Simp:= (q::`=`)-> local e:= factor((lhs-rhs)(q));
    `if`(e::`*`, (remove(f-> is(f<>0), e) assuming complex), e) = 0
:
eq:= diff(v(x),x,x)*exp(x^2) = 0:
Simp(eq);

You could change complex to real if you want.

Like this:

T:= Typesetting:
plots:-textplot(
    [1, 2, T:-mover(T:-msub(T:-mi("u"), T:-mn("1")), T:-mo("&rarr;"), 'mathcolor'= 'red')],
    'align'= {'above', 'right'}, 'font'= ["ARIAL", 'BOLD', 16]
);

You need to change s(e+i) to s*(e+i).

And although it doesn't cause a problem here, please don't use with(linalg).

It can be done like this:

restart:
model:= [x^2*y*alpha[1, 11], x*z^2*alpha[2, 15], y^2*z*alpha[3, 17] + y*z*alpha[3, 8]]:
M:= [
    alpha[i, 0], alpha[i, 1]*x, alpha[i, 2]*y, alpha[i, 3]*z, alpha[i, 4]*x^2, 
    alpha[i, 5]*y*x, alpha[i, 6]*z*x, alpha[i, 7]*y^2, alpha[i, 8]*z*y, alpha[i, 9]*z^2,
    alpha[i, 10]*x^3, alpha[i, 11]*y*x^2, alpha[i, 12]*z*x^2, alpha[i, 13]*y^2*x, 
    alpha[i, 14]*z*y*x, alpha[i, 15]*z^2*x, alpha[i, 16]*y^3, alpha[i, 17]*z*y^2, 
    alpha[i, 18]*z^2*y, alpha[i, 19]*z^3
]:
#Construct new models as a list of lists:
models:= CodeTools:-Usage([
    for i,m in model do
        map(subs, m=~ m+~ subs({`if`}(m::`+`, op(m), m)=~ (), M), model)[]
    od
]);
memory used=41.66KiB, alloc change=0 bytes,
cpu time=0ns, real time=0ns, gc time=0ns
#Lenghty output omitted
nops(%);
                               56

The lines

p2 := with(plots);
inequal(58 < x, x = 0 .. 100, y = 0 .. 2);

should be changed to

with(plots):
p2:= nequal(58 < x, x = 0 .. 100, y = 0 .. 2);

I suppose that vmcofs is very large (millions of numeric entries or tens of thousands of symbolic entries), and you don't need to save every slice? Is that right? 

Create a module that contains a single export, a table:

VM:= module() export vmcofs::table:= table(); end module;

Then every time that you have a slice k that you want to save, instead of doing savelib, do instead

VM:-vmcofs[k]:= vmcofs[k];

When you're done choosing slices, save the whole module with

savelib(VM, lib);

If you change any entries in the array vmcofs[k] after putting that slice into the table but before using savelib, then the table will automatically contain the newly updated slice k, not the original slice k, and only the updated one will be saved to the library. If you don't want this to happen, let me know.

The module is superfluous in this simplistic version; all that's needed is the table. I just included the module because you might find it useful later (for storing other info related to your project). The added overhead for using a module is infinitesimal.

A module with no locals, only exports, is also called a Record. You can create it with the Record command, if you wish. It hardly makes any difference.

Since test is not a reserved word and doesn't contain any special characters, the backquotes don't do anything: test and `test` mean exactly the same thing. So yes, you're right, the help page's usage of backquotes is unnecessarily confusing.

If I do it without Physics Updates, then there's no problem.

The type given as 2nd argument to subsindets can use a specific symbol by referring to it as identical(xi). So, this works:

S:= ex-> subsindets(ex, identical(xi)^integer, e-> H(op(2,e))*e);

Yes, the command is called maplemint.

Great point @sursumCorda ! That is something that I noticed and explained to my students on day 1 of the very first Maple class that I ever taught, some 25 years ago (2nd-semester calculus, freshman students with no Maple experience). Now that you've described the problem, I've thought of an easy solution that Maplesoft could implement: Put a flag on the status bar that indicates which of the following two things has occurred most recently:

1. the worksheet has been edited;
2. the worksheet has been sequentially executed with the Execute-Entire-Worksheet command.

This flag could be as unobtrusive as the single "*" that indicates whether the worksheet is unsaved.

Like this:

eval(expr, int= 1)