Those little tiny bits are on the back side so no worries as long as they don't show on the front of the piece. Personally I don't care for tear away stabilizer so don't use it very often but if it worked for you in this instance then go ahead and use it for the red work. I wouldn't recommend it for the center piece however, you'll need more stabilization than tear away can give for a filled design. For 90% of my embroidery I use cut away stabilizers or Vilene. Sometimes I'll use two layers of Vilene and just treat it as a cut away. I have different weights of cut aways as small designs can use lighter weight and larger denser designs or a heavy weight fabric can call for a heavier weight stabilizer. For light weight fabrics sometimes I'll iron a light weight fusible interfacing to the back. It can be a steep(and expensive) learning curve to figure out what works for YOU. There aren't many hard and fast rules as what works for me might not work for you. The best thing to do is get several types of stabilizer and practice, practice, practice, keeping notes so you know what works for you and what doesn't. Keep your practice pieces so you can refer back to them.
One way to help keep costs down is not to buy the little packages of branded stabilizers. I buy stabilizer on bolts or rolls. It costs more initially but lasts much longer than a small package so is a big saving overall. For instance, you can buy Pellon stabilizers at JoAnns and Walmart on bolts by the yard and cut it yourself. A bolt of tear away for $20 lasts me for over a year, sometimes 2 years. I buy my cut aways on large rolls from Allstitch and it lasts a long time.
There are charts for figuring out what stabilizer to use on many major embroidery sites.
Cari