Strangely enough, the Classic and Standard interfaces use different versions of loglogplot, and in this respect the Classic version is better in Maple 11.  If you prefer Standard, you still have access to the Classic version: try

> `plots/loglogplot`({r1, r2, r3}, 1 .. 10^8, style=point);

Unfortunately, the Standard interface messes up the y axis labels on this plot (which is probably why it was given a different version of loglogplot).

Ah, the problem is that it is Maple 11.  In Maple 12 there are many points plotted in this range.  Here is the plot in "point" style

and in "line" style

It all looks fine to me (except that you really should define, e.g.,  r1 rather than r1(t)).  I think the Maple plots are correct. 

What do you think is wrong with them?

Use the green up arrow to upload your file (preferably a worksheet) and insert a link to it.

The behaviour you described used to be a problem, but I think it has improved considerably in recent releases.  You might be running into numerical problems, but I can't be sure without a look at the actual code.

If you want a "normal" magic square whose entries are the integers 1 to 16, you need more than just a linear system of equations.  Some searching will be necessary...

dharr's method works if you know the intervals where you want the functions plotted.  A more general method is this: 

> Ge1:= y -> piecewise(y >= 1, y, undefined);
  Le1 := y -> piecewise(y <= 1, y, undefined);
  plot([Ge1 @ f, Le1 @ g], 0..3, colour=[red,blue]);

I would avoid the geom3d package here.  For graphing the sphere and the line segment, I would just use sphere and line in the plottools package, putting them together with display from the plots package.  For proving analytically, just write the parametric equations of the line, substitute into the equations of the sphere, and solve.

You might try plotting separately on two intervals and then combining the plots using display.  Thus:

> f:=subs(r = 0.5e-1, D = 0, sigma = .3, K = 100, eq1);
  P1:= plot(f, tau = 0.1e-7 .. 0.01, colour=black):
  P2:= plot(f, tau = 0.01 .. 1/3, colour=black):
  plots[display](P1,P2);

This has come up before.  See e.g.  www.mapleprimes.com/forum/maple-10-doesnt-open-on-a-new-dell-with-windows-xp-sp2 and groups.google.co.ls/group/comp.soft-sys.math.maple/browse_thread/thread/902922b6aa26bfc3.  If none of the suggestions there works, contact Maple support.

Another possible problem is constructions such as (Pi²/8)(x-1)²

If there is no space between the ) and (, Maple treats this as a function call, not implied multiplication: in fact it "simplifies" to (1/64)*Pi(x-1)^4.  This is one reason why I advise new users to stay away from 2-D math input: use 1-D Maple input instead. 
What you want is something like

> plot3d(1 - (Pi^2/8)*(x-1)^2 - (Pi/2)*(x-1)*(y-Pi/2) - (1/2)*(y-Pi/2)^2,
     x=1/2 .. 3/2, y=Pi/2-1/2 .. Pi/2+1/2, axes=box);

Try it with a Maplet, like this:

G:= proc()
   local s,t;
   s:= time();
   Maplets:-Tools:-Set(TB1(value)="");
   while time() < s + 10 do
     t:= time();
     while time() < t+1 do end do:
     Maplets:-Tools:-Set(TB1(append)="a");
   end do;
   Maplets:-Tools:-Set(TB1(value)="Finished");
 end proc;
with(Maplets[Elements]):
 maplet := Maplet([[
    TextBox['TB1'](""),
    Button("Go", Action(Evaluate('function'='G()'))),
    Button("Done", Shutdown())
 ]]):
 Maplets[Display](maplet);

Maple returns int(cos(sin(theta)),theta) unevaluated.  Since this is the "transcendental" case, for which I believe Maple's implementation of the Risch algorithm is pretty much complete, I'm pretty confident that Maple is correct in that this function has no elementary antiderivative.

What do you mean by page limits?

You can insert a page break with Ctrl-Enter or from the Insert menu, and that is visible in the document.  Otherwise, as far as I know, there are no "pages" in a Maple document.  You can use Print Preview in the File menu to see how the document will be divided into pages when printed.

I think this is a matter of automatic simplification, which is pretty difficult to prevent, as well as evaluation.  You could do something like this.

> use `^`=(()->':-`^`(args)') in
    u := 2^0.5;
    eval(map(convert,eval(u,1),rational));
   end use;

                                     u := ^(2, 0.5)

                                           1/2
                                          2

It works with a simple print statement.

Your version will also work in command-line Maple.