I use Pellon brand lightweight fusible interfacing. It says right on the bolt suitable for knits. Most people suggest applying the interface before cutting but I apply it after. I cut my T-shirt blocks and my interface the same size. As soon as my Tshirt is cut I iron on my interface so the knit of the T doesn't have a chance to start curling up. You will avoid the curl and stretch issue if you apply the interface first but that also wastes interface. I prefer not to waste any and applying after I cut has not caused me any undue stress or problems of any significance.
I keep my T-shirt quilts simple by cutting all the same size (12 1/2" to finish at 12" blocks) and connect the blocks using sashing and cornerstones.
http://www.quiltingboard.com/picture...lt-t47094.html
That said, while this style goes together quick and is simple I find them not nearly as visually appealing as T-shirt quilts done with various size blocks.
Batting depends ENTIRELY on how you plan to quilt. I would not attempt to handquilt a T-shirt quilt unless you are doing something like a primitive big stitch. I have tied using highloft polyester fat bat (the pictured link) and I have longarmed using an 80/20 blend.
http://www.quiltingboard.com/picture...ts-t95274.html
I am sure 100% poly or W&N would also work for machine quilting. However, W&N tends to be a heavy batting and add that to a T-shirt quilt and you are talking one very heavy quilt.
Every T-shirt quilt I have made (I have made at least 6) has been pretty heavy due to the weight of the T-shirts so add a heavy cotton batting like W&N and you will have a quilt that weighs quite a bit depending on the size.
Unexpected challenges: Make sure your T-imprint is a silk screen, not the iron on kind of decals that have a very rubbery feel and tend to crack and peel over time. Also use a pressing cloth as the heat from the iron will melt the inks used to make the imprint.
I have never attempted to machine quilt a T-shirt quilt using a domestic sewing machine. I can imagine it would be an enormous challenge due to the weight. You may wish to consider doing some sort of quilt as you go method if you plan to quilt on your domestic machine.