Hobbs has black cotton batting (just google Hobbs batting). You can pre-shrink the batting - it should have directions on the batt package for that.
As a long-arm quilter, I ask for 8" wider and longer then the quilt top for both the batting and the backing. So if your quilt top measured 60" square, I would likethe backing and batting to be 68" square. The quilt backing and batting have a way of creeping during the quilting process. I have had a quilt lose as much as 3" in width and length due to the intensity of the quilting shrinking it.
Do find a long-armer who specializes in doing t-shirt quilts. Some of us have never done them and you don't want us to learn on your quilt. If you would like a recommendation of somewhere to ask about the quilting, just PM me and I'll give you a website - not me or mine, nor am I affiliated in any way, just know they specialize in t-shirt quilts.
Hope that helps and we'd like to see a picture of this when you are finished!