When you bend (or at least when I do) the strings above, or below it will push along with it. That depends on how large the bend is. But, If you feel that the other stings are making noise while you are bending... I would then suggest to bend with your ring finger (or middle) and mute the above string and strings below it with your index finger. You can also mute the above strings with your right hand.
About the string contact with your fingernail... Maybe, you string action is high. I had the same problem with my squier, because I could probably drive a semi-truck under the strings
! Feel free to bring your thumb around the neck and be sure to try "curling" your finger a bit. That could be the issue. I advise you to experiment a little.
Try picking accents as well. By hitting the mute strings above it before you hit the note is not only a cool "flavor" for your bends. But, it also helps you feel a little more comfortable while practicing them.
I hope this was helpful! I don't know if it makes sense, and I hope I got the point that you were asking about. Good luck!