Scale from One End Animation - Reactive Text Box

I see a couple of problems with your basic comp. First, reset the text layer's Anchor Point. Second, delete the Contents/Rectangle 1/Rectangle 1 Path/Position expression. Make sure all of the other Rectangle properties except the Transform Rectangle 1 properties are reset.

Lastly, add this expression to BG Box/Transform/Position to tie

the position of the shape layer to the position of the text layer:

var s = thisComp.layer("This is my layer");
var w = s.sourceRectAtTime().width/2;
var h = s.sourceRectAtTime().height/2;
var l = s.sourceRectAtTime().left;
var t = s.sourceRectAtTime().top;

[w+l,h+t] + s.position

Now, your BG Box layer will stay lined up with the text layer. 

To help with the rest of the problem, try explaining the design goal a little more clearly.  If you are using a Text animator to move the text into position, you can add a time value to measure the text area size at 2 or 3 seconds. If you want to tie the number of characters in the final position to the width of the text box, you can use the start or end values of the Text Animator to modify the text box size so it grows as characters land in their final position. There are too many different design ideas in my head to give you a meaningful solution to your problem without a better understanding of your design.

There is nothing keeping you from using the Background text layer as an Inverted Alpha Matte for the text layer and then turning the layer back on. If you have a Text Animator that moves the text into position in 2 seconds using sourceRectAtTime(2), then the text will not show up until it moves inside the background layer.

Nothing stops you from grouping your shape layer and use the group transform to align things. One way or the other the only thing required is to subtract half the width of the rectangle to accommodate any position changes caused by the size changing.

Mylenium