Let me begin to say that its a really good start. After the first killing frenzy, I was eagerly awaiting the next bunch of robots, only to realize moments later that there was probably nothing more to come. :?

It ran smooth as silk on my system, but I imagine with 3ghz, 1gig ram and a 6800 gt gfx card it better had to

Anyway, on to your question. The whole independant timer thing can be a bit tricky at first, but once you've figured it out, its really easy.

What you gotta do is this:
Create a procedure, called something like processGameEvents. In it you place all the code that draws to the screen or updates your sprites movements. (l'll get back to the movement later)

Then there's a event called OnIdleHandler.
Code:
procedure TForm1.OnIdleHandler(Sender: TObject; var Done: Boolean);
In this procedure you call processGameEvents unless a variable, called something like stopGame becomes true (like a backdoor) so you can terminate the application. Like this :

Code:
procedure TForm1.OnIdleHandler(Sender: TObject; var Done: Boolean);
begin
   processGameEvents ;

   if stopGame then form1.Close;
   done:=false;
end;
Lastly in the oncreate event set Application.OnIdle := OnIdleHandler;

Thats all there is to it.

Now about movement. If you do something like sprite.x := sprite.x + 2, which normally is a good value when using the dxtimer component, your sprites will now suddenly move insanely fast.

In order to let an object move at the correct speed each frame, you have to calculate the time between frames. (in the processGameEvents procedure)
Code:
NewTime := TimeGetTime;
UpdateSpeed := (NewTime - OldTime) * OneMSec; { get the speed to update objects }
OldTime := NewTime; { store this value for the next update }
Together with the UpdateSpeed you can now update the sprites, something like
Code:
sprite.x := sprite.x + (0.2 * UpdateSpeed)
I hope this helps you a bit. But in case it doesn't, have a look at the source from Pop the Balloon, on my website. You can find all this in there, be it with slightly different variable names.