Hi, you and your family have done a wonderful thing, I applaud you and them.
I used to have a heck of a time with binding too, now I hand baste the edge of the quilt, close to the edge so it will be covered by the binding, no more puckers!!! The corners, I stitch all the way threw then take out enough stitches to make a nice corner, it seems to vary with binding size but when you get the first one done you can guess where to stop with the other 3, just don't lock stitch. I sure hope this helps you with binding, it did me.