Can Nasal Bridge Cause Breathing Problems Too?

You have a septal deviation. Sounds like you may also have nasal mucosal congestion. This causes variable and alternating congestion. It's from swelling of the fleshy parts of the nose called turbinates.

Generally, the nasal bridge is not a major contributing factor to problems with nasal breathing.
Deviated septum is the most common cause of nasal congestion. Enlarged turbinates is the second most common cause of difficult breathing difficulty, and usually associated with a deviated septum. Other structural problems with the nose, such as small or weak nostrils, scarring, narrow or collapsed nasal valve may also contribute to nasal obstruction.
Other conditions, such as allergies, medication side effects, chronic sinusitis, irritants, and systemic illnesses may contribute to nasal breathing issues. These non-structural conditions can be treated with medicine and may not require nasal surgery, such as septoplasty or rhinoplasty.
Speak with a rhinoplasty surgeon to help identify and correct such problems. Only after a comprehensive approach will he/she will be able to determine the most appropriate solution for you. Best of luck.

Your nasal breathing issues are most likely associated with your septal deviation.  The fact that you have consistent obstruction on one side verses the other is a critical piece of information.  If this changed throughout the day or night then a separate issue of turbinate hypertrophy (humidifiers in the nose) may also exist.  The dripping of the nose is likely related to some mild environmental or specific allergy that creates more drainage.  If you have not undergone a formal allergy consult you may want to consider this.  A Netti pot may be a simple and easy natural solution for this issue.  The pain and discomfort on the nose in cold environments is impossible for you to control but unrelated to the deformity.  It truly is a reaction of your nasal lining to a change in temperature.  By maximally hydrating your nose ... with a Netti pot type option you may find this will improve...but do not expect surgery to solve this problem.  If a unilateral issue exists only then you will likely find that a septoplasty significantly improves your condition.
 

The deviated septum, about which everyone has heard, can obstruct an airway.  Bu if the valves, which stabilize the sides of the nose, are weak and collapse on inspiration, the airway will be poor even if the septum is straight. My published airway research shows that the valves are even more important than the septum in supporting the airway.   Make sure that your surgeon evaluates them.

The turbinates, which warm and humidify the air and enlarge when the airway is bad or because of allergies, can also block your breathing.

When airflow is turbulent and the airway is blocked, the nose can run.  Patients who have bad airways also blow their noses a lot, which just makes them run more.  Once your surgery is healed, these problems will improve greatly.