TCA Peels Before Fraxel Laser to Get Full Benefit?

"Shaving down the sides of the nose" is not a clear statement. Doing a septoplasty and turbinate reduction is for improving the nasal airway and is not a cosmetic operation. Doing procedures to change the external appearance of the nose (Rhinoplasty) is a different operation that may or may not be done at the same time as the septoplasty. It needs to be clear if you are asking for cosmetic surgery , and if so, what you want changed such as thinning or reducing the dorsum, refining the tip, making the nose smaller, etc.
Make sure you are clear on the plan before agreeing to surgery.

A. Fraxel and TCA peels: I cannot see the need for that many TCA peels prior to a series of fraxel laser treatments. Sometimes I do combine other laser treatments with Fraxel in order to enhance results. Some examples of lasers used in combination with Fraxel are:

Vbeam pulsed dye or Gentle YAG 1064 for bood vessels or redness
GentleLase long pulse alexandrite or a q-switched alexandrite laser (ex: alex trivantage) to treat pigmented spots - freckles, melasma
Smoothbeam non-ablative laser for depressed scars

Additionally, I commonly use 90% TCA to the base of ice pick type acne scars (CROSS technique) immediately prior to Fraxel to help with these difficult to treat deep scars.
B. Darkness after Fraxel: Even with olive skin, the risk of darkness or what is called medically PIH - post-inflammatory hyperpigmentation - is very rare even after moderately aggressive Fraxel treatments. If you aer concerned about PIH or have experienced darkening after skin wounding previously, then discuss with your physician adjusting the settings of Fraxel laser to minimize the risk of PIH. PIH is also very treatable if it does occur.