See the help page ?plot,devices for the supported file formats.  The ones relevant to you are dxf and perhaps pov.

I don't get a crash or freeze (in Maple 14.01, Classic or Standard, under Windows 7).  However, in Standard I do get a popup window after a few seconds, with heading "Error in Shutdown" and text "The element _Maplets_reference_1 does not exist".
After clicking OK, Maple functions normally.  There may be a bug here. 

With A as the Matrix you want differentiated with respect to t, use either

map(diff, A, t)

or

diff~(A,t)

Why be satisfied with such a crude approximation?  Take the equation in polar coordinates, differentiate implicitly, and numerically solve a system of differential equations for r(theta) and A(theta), where A(theta) is the area of the sector of your region from angle 0 to theta.  The answer wil be A(2*Pi).

This is a nonlinear autonomous second-order DE.  The general solution is unlikely to be expressible in closed form.  Maple does the best it can, "reducing" it to a first-order DE for u' in terms of u.  This may not be helpful to you.  So what can you do?
Given initial conditions and values for the parameters alpha and beta, you can get numerical solutions with dsolve(..., numeric).
Or you can get series solutions with dsolve(..., series).  Or (again given values for the parameters) you can plot a phase portrait and solution curves with DEtools[DEplot].

with(Maplets[Elements]):
 maplet:= Maplet(FileDialog[FD](filefilter="mw", filterdescription="Maple Worksheets",
    directory="d:/test",onapprove=Shutdown([FD]), oncancel = Shutdown())):
 Res:=Maplets:-Display(maplet):
 if Res <> NULL then 
   Worksheet[Display](Worksheet[ReadFile](op(Res)));
 end if;

If exporting from Classic (via plotsetup), use cps rather than ps as the plot device.  It doesn't support typeset subscripts, though.  And (in Maple 14.01 Classic) after

plotsetup(maplet); plot(y[r]^2,labels=[y[r],x[c]]);

although the resulting Maplet did show subscripted labels, trying to export the result as EPS from the maplet's File menu I got the error

AXESLABELS must contain two NAMEs or STRINGs

You can display an Array of plots.  For example:

> plots[display](Array([[plot(x^2,x=-1..1),plot(x^3,x=-1..1)],[plot(x^4,x=-1..1),plot(x^5,x=-1..1)]]));

Flatten is in the ListTools package; randmatrix is in the deprecated linalg package, but it's better to use RandomMatrix from the LinearAlgebra package (and, btw, Vector and Matrix rather than vector and matrix).  So:

> with(combinat);
 with(Statistics);
 M := LinearAlgebra:-RandomMatrix(5, 5);
 XYValues := convert(sort(permute([seq(1 .. 5), seq(1 .. 5)], 2)), Matrix);
 ZValues := convert(ListTools:-Flatten(convert(M, listlist)), Vector);
 Fit(a*x^2+b*y^2+c*x+d*y+e, XYValues, ZValues, [x, y]);

My favourite probability text is KL Chung, "A Course in Probability Theory",
<http://books.google.ca/books?id=zu80w7wd40UC>.  You might also try Grimmett and Stirzaker, "Probability and random processes", <http://books.google.ca/books?id=G3ig-0M4wSIC>.

Hint: the centre of the circle, the midpoint of the chord and one of the endpoints of the chord form a right triangle.  What does Pythagoras tell you?

This certainly shouldn't happen.  There may be something wrong with your installation of Maple.  You should contact Maplesoft technical support.

See Jacobian in the VectorCalculus package.

> S:=solve({cos(x)=0},AllSolutions);
   X:= subs(S,x);
   isolve({X>=-Pi,X<=Pi});
   map(subs,{%},X);

Are you asking the C program to do symbolic algebra (e.g. taking the Jacobian of a vector of more-or-less arbitrary expressions)?  Then perhaps you can use the OpenMaple API so your program can call Maple to do this.  Or are your symbolic expressions known beforehand (apart from parameter values)?  In that case, you want to get Maple to do the symbolic work beforehand, ending with a collection of expressions that your C program will just have to evaluate numerically when given values of the parameters.  The CodeGeneration package can help with that part.