The Dodge and Disengage actions are primarily considered as an option to take as your primary Action (PHB p.192):

Dodge

When you take the Dodge action, you focus entirely on avoiding attacks. Until the start of your next turn, any attack roll made against you has disadvantage if you can see the attacker, and you make Dexterity saving throws with advantage.

Disengage

If you take the Disengage action, your movement doesn't provoke opportunity attacks for the rest of the turn.

There are some classes however, that allow you to "Dodge", but these class features aren't the same as the Dodge action. For example, the Rogue's Uncanny Dodge feature halves damage, as a reaction (PHB p.96):

Starting at 5th level, when an attacker that you can see hits you with an attack, you can use your reaction to halve the attack's damage against you.

The Monk does gain a feature to allow them to Dodge as a bonus action, at the cost of a Ki Point (PHB p.78):

Patient Defense

You can spend 1 ki point to take the Dodge action as a bonus action on your turn

And then a feature similar to the Rogue's Uncanny Dodge (PHB p.78):

Starting at 3rd level, you can use your reaction to deflect or catch the missile when you are hit by a ranged weapon attack.

Outside of this, there is very little that allows a player to make these actions as a reaction, or a bonus action.

The only other suggestion I might make is that Matt Mercer's Critical Role series allowed players to use their Bonus Action as "another Action", to basically make things more cinematic. This was generally a spur of the moment decision to keep things moving, rather than to stop the game and check the rules, however.

You have it almost right

The only other thing to consider is the Barbarian's reaction, movement, and free object interaction. If they can attack a hostile creature with any of those or take damage as a result of any of those they can also ensure their rage doesn't end on the first turn.

Generally, the most common way that would happen is for the Barbarian to provoke an attack of opportunity and that attack to be made successfully and result in some damage, but many other options exist; for example, the barbarian could deliberately step on a caltrop or other hazard.

All of those ways, except perhaps getting to use your reaction to attack turning the same turn, are almost always bad and unnecessary-- the Barbarian should just wait until they take damage to super rage instead. Reaction based attacks currently at-best require you to be attacked and most require you to be hit, so those are kind of a bad idea, too, unless you have built around it. Commander's Strike is an exception, if an allied battle master fighter goes before you in the first round of combat and you are not surprised and you are in position to make an attack (it doesn't have to be a melee attack so that last part isn't unreasonable) and the GM rules you can take reactions before your first turn in combat, but it is bad for a battlemaster to use, generally, because it takes both a bonus action and an attack and your reaction and that is a lot of stuff so it's 1) unlikely your allied battlemaster fighter has this maneuver and 2) unlikely it's better for them to use it even if they do have it than to do two additional attack-equivalent thingies and leave you with your reaction available.

Pretty much the only scenarios I can come up with where you'd want to do this are where you are involved in a decent-sized melee and there's an AOE trap you can trigger to hit you and the enemies but not your party and you want to benefit from your resistance to damage and where you have an allied battlemaster fighter as a result of having used a Deck of Many Things and he or she stands in the back with a longbow and uses Commander's Strike instead of attacking.

So, in essence, yes, your reading is correct, but technically no, and the answer might change as better ways to use your reaction to attack during your turn are published.