While riding a flying steed, can a character take the Dodge action? Does it matter if the mount is being controlled or not? It seems weird that a person can "dodge" while sitting down.
The character can take the Dodge action because Dodge is one of the listed actions all creatures can take.
The mount can take the Dodge action because a) if it is controlled, Dodge is one of the permitted actions and b) if it is independent, Dodge is one of the actions all creatures can take.
You don't need rules to let you do stuff like this
You need rules to tell you you can't. Whenever the rules say "You can..." it means you can do that thing all the time whenever no matter what, unless something else specifically says it stops you.
Sometimes what stops you is the DM not liking the result of using the rules, rather than something within the rules themselves, and sometimes people call that 'common sense' so as to avoid discussing it. But common sense is not something we can advise you about with certainty when you ask about how a rule works; common sense is in many ways a reflection of what your DM believes about the metaphysics of real-life and DMs do not, in general, subscribe to a single metaphysical cosmology consistent across the whole RPG playing community.
Furthermore, even should your DM decide that, for example, the peasant railgun doesn't work because actions are supposed to be simultaneous or something, that doesn't make that an actual part of the game-system or rules. If we're talking about how the rules work as concerns passing objects, limits your DM might have come up with like 'it can't go farther than you can run in 6 seconds' or whatnot aren't a part of that; the rules still allow the peasant railgun, only your table doesn't.
So, then, it should be clear from the text related to mounted combat that nothing about riding an animal affects your ability to dodge. The same is true for flying. There's no text, really, about riding flying animals, and what little there is certainly doesn't talk about not being able to dodge.
Therefore, since the rules say you can spend your reaction to dodge, you can do so, just like you could in any other situation that doesn't specifically and explicitly forbid you from doing so.