If you square as you go along, you can usually tell whether things are lining up right or not. If I am doing borders, I like to square before and check after, by measuring several places along each side and adjusting/cutting as needed. My final check is measuring both ways diagonally, and if they are the same size (or pretty close) I go forward with assembly.
The eyelet should definitely be lined. Today's battings don't shed as much as older kinds, but it will be worth it to at least use an iron on interfacing. I'm sure you could get some interesting effects by using a contrasting color that showed through the openings.
Starting quilting from the middle gives you room to smooth out any imperfections. If you baste well, I wouldn't think where you actually started quilting would be that big a deal. I learned to do an X and + years ago for handwork, but that was before spray glue and great battings.