The answer is simply no, a caudal septal extension graft does not need to be stabilized with extended spreader grafts. They can simply be secured to the lower (caudal) part of the pre-existing septum using stable stitches. That effectively elongates the septum, pushes the lower lateral cartilage down to derotate the tip.
An extended spreader graft is a second way to lengthen or turn the nose down. My fellowship director, Dr Wally Dyer, published this as the DART, or Dyer Advancement Rotation Technique, many years ago. In his description, the extended spreaders were secured to a columellar strut, which then served as an anchor to turn the tip down. There have been a number of experienced rhinoplasty specialists who have described similar techniques.
Either way surgeons have their preferred way to move the tip downward while still controlling projection. A caudal septal extension graft is one way, extended spreader grafts represent another option, and other tip grafting techniques add yet other steps. Trust your surgeon to work with the technique(s) that he/she feels most comfortable in order to serve your needs.
You always want to consider chin implants, nonsurgical chin augmentation, submental liposuction, or kybella. How each of these relate to your anatomy is beyond the scope of the question, but can be answered by your surgeon at the time of a consult. At that time they can use your profile picture and image modification software to discuss what each change means to your face. My initial impression based on your picture is to consider the submentum first, meaning lipo or kybella.
You can indeed bring up your bridge with soft tissue fillers. To a degree. If you try to build it up too far or too fast, you will lose precision and the nose just looks like a swollen blob. So serial, stepwise, conservative, thoughtful augmentation might get you some of what you are looking for. The problem can also be that the tip does not respond well to filler. Much less likely to get precise highlights or projection, so temper your expectations here. The key is to first find a surgeon to do this. I would not suggest letting anyone who does not operate on a nose consider a filler here. Second you have to have an honest conversation with that surgeon about your expectations, vs what can be achieved. If you are on the same page, proceed with anticipation!
There are many options for rhinoplasty that will open up the nasal airway. They include septal straightening, turbinate reduction, internal valve and/or external valve collapse correction. Unfortunately there is no way to answer your questions without an evaluation. You may have other reasons for airway obstruction, like mucosal congestion, that would not change with any surgery. So a global answer that promises airway improvement with rhinoplasty is premature.
Patients with significant allergies, non-allergic rhinitis, or even a few congenitally-narrowed nose obstructions would still have to work with medicines and non-surgical treatments.