Will Columella Drop in Place After Rhinoplasty?

It is very hard to determine specific issues related to another surgeon's work, especially without photos from before and after and an operative report. One question I would ask is if this a new deviation or a partially corrected/uncorrected persistent one? While I do not recommend pushing the nose as you describe, you should follow the advice of the surgeon that perfomed the work. Perhaps there is some technique he or she used that I am not familiar with...
In many instances, a suture will be placed from the bottom of the septum to the columella. This suture is typically a dissolving one. Depending on what suture was placed, it may take as much as 3-4 weeks to go away. Also, the incision at the bottom of the septum tends to make the area stiff for a while after surgery. Other possibilites are that the tip was sewn to the septum to help change its position. Finally, if an open approach was used, the columellar incision may take a while to soften. Fortunately, the issue you describe does usually correct over time.
Try not to worry and listen to your surgeon's advice. He or she knows best how to take care of you in the early post operative period. If problems persist down the line, there are always options for repair.
Best of luck.

Without knowing the timing of your surgery and techniques utilized, I can generally tell you that the nose will settle and the columella may likely become less elevated with time as the scar tissue softens as stitches absorb. Give it at least 6 months and preferable 1-2 years before making any final judgements.

 Nasal exercises are not something that I find very common. Without seeing your photos or examining you in person, I could not give you any suggestions other than to stick with your doctor during the post-op period.  If you are not happy you can always get it revised down the road.