A long delayed answer, but one that might be useful for people.
There are only a few things you can do regarding fermentation speed, but it may result in other factors.
Yeast strain. Some yeasts act faster than others. However, each strain has their own flavor profile. (I'd consider guessing turbo yeasts might work, but they'd be more expensive and aren't as common to find. There's a strain that should be able to eat about 6kg of sugar within 24 hours. Another problem would be is that you probably have to keep an extremely close eye at this one and chill it fast, and because there's still suspended yeast, it'd be very yeasty.)
Temperature. Different strains work best in different temperature conditions. Higher temperatures generally means faster. But, yeasts at higher temperatures put off different flavors which can be detrimental to your desired soda flavor.
Nutrients. It's possible that you can add some nutrients for healthier yeast which might be able to reproduce and consume sugar faster.
Pre-hydration. Rehydrate the yeast prior to adding it to the soda.
Quantity. Not sure about this one, but you could hydrate more yeast so it doesn't populate as much. (maybe)
The yeast flavor comes from either yeast or yeast byproduct. Probably yeast. What happens is that yeast is all around in the liquid. Chilling for a long period of time or clarifiers makes them drop out of suspension. You could consider using irish moss when you make rootbeer to act as a clarifier, and also to possibly add to the froth/head of the root beer.
Secondary: GF Czech Lager
Waiting to be kegged, Italian Primitivo
Kegged&Ready: GF Orange&Coriander, GF Honey Lager, GF chocolate ale, GF English ale, Island mist (zinfandel), Island mist (cbry malbec).
Bottled: Infected Mead, Dry Hard ciders, Accidental Sorghumwine, various unnamed.