Is This a Proper Fit? Or Should the Crown Be Far Enough Down That Flossing Doesn't Pick Up on the Crown's Edge?

See your dentist and have them take an X-ray.  It is probably cement left under the crown that can be removed fairly easy.  If no cement is found, it may be a bulky margin and your dentist can evaluate for the proper means of treatment.  Worst case scenario is that you would have the dental crown remade.  

What you describe is called an overhanging margin. This is not acceptable dentistry and should be remade. The shelf created by the misfitting crown is a food and plaque trap which leads directly to decay and disease, bone loss and tooth loss over time. Now is the time to fix that, before it creates an undesirable outcome.

There should not be an edge that catches floss.  Sometimes that is just a little excess dental cement that the dentist can smooth off.  If the crown doesn't completely fit the tooth and there is a gap, that is also a reason why your floss catches.

I suggest you return to your dentist, tell him your concerns.  He can evaluate the crown he did for you, probably take an x-ray.  Sometimes these problems are only caught on x-rays, and most dentists don't take x-rays immediately after placing new crowns because of the hypersensitivity of many people to the cost and thought of "too many or unnecessary x-rays". 

The last thing your dentist would want would be an improperly fitting crown that might later give you some problems.

BTW, congratulations on being such a great flosser.  That is the best thing you can do to insure you will keep your gums healthy and prevent unnecessary cavities.