Once the cast is off, is it likely to drop down and be a normal nose?

Yes once the cast comes off the nose will drop and look more normal, the swelling will also start to go away at that time. 

 It truly takes 1 full year for the entire dorsum (profile) and sidewalls of the nose to fully heal and for all the swelling to dissipate. Rhinoplasty healing is a tedious process because it is one of the only surgeries where we are not removing excess skin. It takes 3 years for the tip to fully heal (only if significant tip work is performed). The tip takes the longest time to heal and mold to the new framework because it is the thickest skin of the nose. The skin over the profile is thinner than the skin over the tip. Patience is required during this healing phase. Great idea to talk to your surgeon about your concerns but keep in mind healing takes time.

I am glad that you are addressing your concerns with your Surgeon, which is ultimately the best way to handle these situations.  However, you can expect your nasal tip position to migrate downward over time.  One of the most common concerns patients have immediately post-operatively is elevated tip position. This is usually by design, expecting later downward rotation.  Remember that it takes a year for swelling to resolve after rhinoplasty.