Flavor is very subtle, but it’s there. In cheap vodka it’s present because it’s produced as cheaply as possible and has impurities that could be filtered out; higher end vodka brand curate a “signature” flavor. It’s obviously way less flavor than most other liquor, but even high end vodkas taste slightly different.

From column stills, yes — it’s ~95% ethanol out of the still that’s watered down. With a pot still, it comes out as 60-80% ethanol. Majority of modern vodka production is in column stills (pot stills are more common with whiskey). With a column still, there’s some flavor hiding in that 5%. Even more room to hide flavor if it comes from a pot still.

Base ingredient is usually grain or potato and it does make a subtle difference.

The water makes a difference since it’s 60% of the final product. You don’t want vodka that’s 60% Dasani (garbage water).

Filtration process/aggressivenes makes a noticeable difference. Heck, you can significantly improve the cheapest bottom shelf vodka at home just by running it through a Brita filter a few times.