A Quilter's Requirement?

I feel like it's a requirement as a quilter to do at least one t-shirt quilt in your "career."  This is pretty perfect for me too; my whole life my mother has been telling me to get rid of my t-shirts.  Many of my t-shirts (much like my baby blanket) are worn out and were loved for a few too many years.  It turns out that once I start making stuff for myself I can't stop-- plus, I'm an Aries and we are really good at starting new projects, but very bad at finishing them...

I wasn't really sure how to go about making a t-shirt quilt for myself for many reasons.  First off, I don't have enough of any particular type of t-shirt to do a full quilt out of one type (I am very close with gymnastics t-shirts, but some of my oldest have mysteriously disappeared and there are some that I still wear to the gym).  Also, it would be hard to do the same sized squares for each t-shirts because they range in size from a kid's medium to a men's XL.

I decided to do a little online research to see some of the quilts that others had done, and I came across this pattern of sorts.

It is definitely too big for what I want 1) I want a lap quilt, not something you could put on a king sized bed and 2) I don't have the many t-shirts that I'm willing to cut up, so I printed it out and cut and taped it until I thought that it looked good and doable.

I have begun to cut the t-shirts, which is proving quite hard.  I hate squares to begin with, but with no salvage and with 15 year old jersey knits, some of them look a little wonky.  I'm going to have to get some interfacing if I actually ever want to finish it.  An other problem that I am encountering is that many of these t-shirts were retired because they were either so worn out that they have holes in them or I spilled something and stained them, or I used them in grad school as painting shirts.  For these reasons, cutting them is even trickier.  Obviously, I cannot have any holes in the quilt and I'm trying to keep it so there is also no paint, or food or hair dye stains.  I'm clumsy.

On a side note, does anyone know if you can recycle cotton?  I don't want to throw of the fabric that I cut off of the t-shirts and I can't imagine choosing to work with jersey.