If you can get a good balanced stitch so the bobbin thread doesn't show on top, it isn't necessary. Maybe a better choice would be to match your bobbin thread to your embroidery design. Or just loosen the bobbin tension a little so none of the thread pulls to the top. Are you doing light open designs or big filled designs?

Sorry I cannot help, my quilting knowledge is 0 but we have lots of quilters here who I am sure would be able to help. Good luck.

I wouldn't use black stabilizer or black thread. It will be hard enough for the lighter thread to show or cover black fabric without adding more blackness. Also black thread has a greater tendancy to break.

Depending on the density of your fabric the stabilizer might show through and it might not. Hold the white stabilizer behind the black and see what it looks like. If it doesn't show then it's OK. If it changes the look a little that's OK too as long as you use the same stabilizer throughout the whole project.
For the thread I would use black unless your test shows that absolutely no bobbin thread comes to the top. That's assuming that you want your design to blend in with the fabric. I've never heard of black thread breaking more than white but I don't know everything.

I've never had any issue with bobbin thread showing through so I have never matched the bobbin thread. If you are using tear away stabilizer, there will probably be little stabilizer left to worry about. Let us know how your test stitch works out. Good luck with your quilt.