I wonder if your needle is hitting the bobbin or throat plate or something else that would cause the needle to bend.

A popping noise usually means the needle is dull & needs to be changed. However, if you have been trying other needles, that couldn't be the reason. Hope someone can help you. May want to try a "sharp" or "top stitch" needle.

Does it make the same popping noise if you were to sew normally through the same fiber sandwich and not free motion?

If your top thread is showing on the bottom you have a tension issue. You need to tighten the top tension or loosen the bottom bobbin tension...this may be causing the "popping" noise...also check the space between the foot and the quilt sandwich...it sounds a bit like your fabric sandwich might be bouncing a bit. I do not think it is the needle that is the problem. I have also heard this popping sound sometimes when free motion quilting and it is usually the sound of the needle going through "stiff" fabric which could be because of your starch. But you definitely need to adjust your tension...also, do you hear this sound when just stitching? If you have a walking foot, try the same quilt sandwich with the walking foot..do you still hear the popping?

Have you successfully free motion quilted before on your 7700? If so can you remember what is different. I keep a notebook with settings used on different projects so I can refer back to later if I need to.

You already have some great advice but I will give a couple of suggestions also.
Have you tried a different brand of needle? I had several bad batches at once from a good brand. Most had burrs and made that popping noise.

Are you switching to single needle hole plate? It may have a burr inside that hole that needs filing.

Other than that it could simply be the fabric combination. Sometimes fabric weaves are quite fine and they don't like to play nicely when Fmq.
There is a yahoo group for the 7700 and they may have a solution for you if suggestions from everyone here do not help.

I have the janome 6600 and even with the fmq blue dot bobbin case and countless different settings, needles and thread it still won't behave properly. I usually quilt on my brother pq1500 as it fmq perfectly, I can even free motion pieces of lace together. So I no longer waste time trying to get my 6600 to behave while I have a machine that does fmq well.