According to Google AI, Garmin's VO2 Max estimation method doesn't use BMI or body fat percentage, only weight so the answer to that part of your question would appear to be no. It also says that because the formula assumes weight is a factor, it can be less accurate for individuals with high muscle mass (high BMI, low fat) as it may treat that weight as "excess" weight, leading to an artificially lower VO2 max score.
It also says that newer Garmin devices specifically use BMI or body fat percentage to calculate a "Fitness Age" metric, so the answer to that part of your question would appear to be yes.
Neither weight nor BMI are used in the VO2max calculation. The estimate is simply based on known correlation between speed and heart rate to the oxygen intake. Bigger or heavier persons do consume more oxygen, but since VO2max is a relative value per kilogram of body weight, for the estimation, the weight, and also the BMI or fat%, are not needed.
Read about the method Garmin uses, at Microsoft Word - EE-white-paper_1 7 6_modified_corrected 2.doc
In brief, it works on the same principle as Cooper test, with the difference, that it extrapolates submaximal HR values, and that it exclude parts of the run that would skew the results. BTW, you do not need the weight, BMI or fat% for the Copper test either.
As for the Fitness Age - as PubBike wrote, newer Garmin devices indeed use Body Fat% (if available) or BMI (if you have no fat% in your stats) for the estimates. The list of devices using BMI/Fat% can be found here: What Does Fitness Age Mean in Garmin Connect? | Garmin Customer Support