How would you go about allowing moving objects on the screen with the interrupt running?  Would you have to somehow sync the refresh rate with the amount of frames? (Sorry if I worded that incorrectly)

You don't necessarily have to sync with the interrupts, but it would be good practice to use it to keep a constant framerate.  
The Stop command halts the CPU until an interrupt occurs.  Remember that a frame is not drawn every interrupt in this case.  You could call a piece of code each time the screen is drawn, but you have to keep your code short to avoid interrupting on itself.  Instead, you can set a variable each time the screen is drawn, then have a piece of main code that waits for the variable to be reset.  

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I wrote the following code to have more precision in screen drawing.  The 8.16 fixed point number allows the screen display rate to alternate between whole numbers.  While you will be able to make the screen better, it will still not be as clean as the faster crystal timer interrupts.

:.AXEGRAY
:
:ClrDrawrr
:Rect(48,0,48,64)
:Rect(24,0,24,64)r
:Rect(72,0,24,64)r
:
:1→G
:0→H
:9→S
:0→T
:FnInt(DG,0)
:Fix 0
:Fix 5
:
:While 1
:If getKey(4)
:If T>(T+256→T)
:S+1→S
:End
:End
:If getKey(3)
:T+1→T
:If T=0
:S+1→S
:End
:End
:If getKey(1)
:If T<(T-256→T)
:S-1→S
:End
:End
:If getKey(2)
:T-1→T
:If T=EFFFF
:S-1→S
:End
:End
:Text(0,0,S►Hex)
:Text(16,0,T►Hex)
:Pause 100
:EndIf getKey(15)
:LnReg
:Fix 4
:Return
:
:Lbl DG
:G-1→G
:If G=0
:DispGraphrr
:If H<(H+T→H)
:G+1→G
:End
:G+S→G
:End
:Return
EDIT: Wait... This code crashes in Wabbit.  It works fine on my calculator, though.

There is almost absolutely no chance that the interrupt occurs between erase and draw

You could also temporarily disable interrupts before the erase and enable them after the draw.  That way, you will never display a half-drawn screen.

EDIT: fixed quote formatting

Also I've compared 3 vs 4 levels and 3 looks clean when scrolling while 4 looks really blurry. Maybe it's because I can't calibrate it correctly.

What you are seing is most likely an artifact of the grayscale mask.  How fast are you scrolling in relationship to the interrupts?  Have you tried scrolling at faster or slower rates?