Some of the lambda's in vvalue were not formed with the double underline (use lprint vvalue to see this). After fixing this, simplify(...,size) executes in a reasonable time.
 

Download sol1det.mw

Hard to tell without an uploaded worksheet (use green up-arrow), but it looks like NN and MM weren't given values.

To evaluate y at a particular x value use eval (for evaluate). For example,

y:=0.1*x+0.3*x^2;
eval(y,x=2);

gives 1.4

To do a lot of them, set up y as a function (technically a procedure), and put the x values you want in a list:

y:=x->0.1*x+0.3*x^2;
xvals:=[seq(i,i=0..2,0.5)];
y~(xvals)

gives [0., 0.125, 0.400, 0.825, 1.400]

plottools:-exportplot can export .svg (should also do .tiff but I got an error message). The .svg file looks correct in CorelDraw.

Download exportplot.mw

From the ?dsolve,numeric help page "All IVP methods can be used for complex-valued IVPs with a real-valued independent variable". Not sure what it would mean for a complex independent variable - some sort of path through the complex plane would be needed.

@dharr CayleyTableGroup is another way. (CustomGroup could also be used.) - works for add and multiply

Download mod7.mw

I'm assuming you really want a group, rather than just doing arithmetic mod n. The addition case is just the CyclicGroup:

Download mod7.mw

Following @Mariusz Iwaniuk but can't get the last step.

Download summation.mw

I clicked on a pMML link (5.13.E1), which downloaded a file. I opened it in a text editor, selected all the text and copied it. In a Maple 2D input region, I pasted and it asked me if I wanted to convert MathML to 2D input, and I said yes. It looks a little strange in the input region (not as bad as here), but seems to be functionally OK.

Download PMML.mw

Perhaps you want to assume n is an integer, or even or odd, in which case the following may help. If you want to use the +/- symbol, then that is not as easily done.

Download Xn.mw

@Kitonum's method requires retyping of the variable names on the lhs. This can be avoided with assign.

Download solveEqs.mw

The ExportMatrix command can take a list of Matrices and write them to a MATLAB format file. See the help page for more details.

I don't know the exact answer. I found it confusing that you added the types in the module within the module; maybe it is a scoping issue. Why not just add them in the ModuleLoad procedure of RationalTrigonometry? (And for debugging purposes, I'd comment out any lines that remove them, in case they were added and the removed before you intended.)

Perhaps this is what you recall.

de:=gfun:-algeqtodiffeq(y = y^2*z + 1, y(z))

gives

The thing on the left-hand side of an assignment (:=) can't be an expression like lambda^m. You could assign to, say, lambda_m, but it would just be a name.

Perhaps you want something like eqn[m] instead of lambda^m?