Your RAW interpretation is correct

The interaction with lesser restoration

When casting lesser restoration on a target affected by feign death, the RAW result would be the same as casting it on a target affected by the likes of blindness/deafness, contagion, sunbeam, or any other spell imposing the blinded condition: the blindness is ended.

This is odd, though, because with feign death the target is in a state of blind and immobile catalepsy verging on death because they opted into it, not because they were harmed. My guess is that the designers didn't consider that anybody would bother to cast lesser restoration on the same target (or didn't care). The RAW leads to some narrative dissonance that I suspect isn't intended, which I'll address in a later section.

The interaction with freedom of movement

When casting freedom of movement on a target affected by feign death, there's no condition being ended. Rather, freedom of movement effectively suspends one of the effects of feign death.

Here's the relevant text from freedom of movement:

For the duration, ...spells and other magical effects [can't] reduce the target's speed...

And from feign death:

For the spell’s duration... [the target's] speed drops to 0.

In this case, we need to consult the rule for overlapping spell effects to figure out how they should interact (see Combining Magical Effects from the Spellcasting section of the Basic Rules):

The effects of different spells add together while the durations of those spells overlap.

So the RAW result is that the effects add together. But what does it mean to "add together" two directly contradictory effects? Since they're mutually exclusive, one spell has to supersede the other (or they have to cancel out, thereby imposing no movement adjustments on the target, which in this case is equivalent to freedom of movement's benefit anyway). So which one should the DM choose? Well, the entire point of freedom of movement is exactly to counteract effects like this!

Once again, by RAW, the result is that the effect of feign death should be superseded. Yet again the RAW leads to some narrative dissonance that I suspect isn't intended, so let's address that now.

How to resolve the dissonance

I have no evidence that the RAW result isn't intended. However, you're asking the question because, like me, it just feels like it shouldn't work this way. As you said in your question, "this doesn't quite seem like it's the intent of the spell." So let's operate under that assumption: that there's some dissonance to work through.

At my table, feign death would supersede the other spells, because my gut tells me that when you opt into its effects you don't get to have your cake and eat it too. At your table, you're free to use the RAW if you think it promotes more interesting gameplay or narrative (magical espionage action: "huh? what's that? it's just a body"), or you're free to share my gut feeling and declare that feign death supersedes. In any case, it's always up to the DM to run the game in a way that makes sense for them and their table, so you should do that.

My suggestion is to err on the side of the ruling that seems most consistent in your game's world and for the tone of the story you're trying to create. What works better for you: that a cataleptic creature indistinguishable from dead can see and move or that it can't? Your answer to that question will solve your problem.

A Prone Creature can Dodge

Nothing in the rules specifies that a creature cannot take the Dodge action while under the effects of the Prone condition.

The Prone condition (PHB, p. 292) states:

• A prone creature's only movement is to crawl, unless it stands up and thereby ends the condition.
• The creature has disadvantage on attack rolls.
• An attack roll against the creature has advantage if the attacker is within 5 feet of the creature. Otherwise, the attack roll has disadvantage.

And the Dodge action (PHB, p. 192) states:

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. You lose this benefit if you are incapacitated or if your speed drops to 0.

What would happen in this scenario is that any melee attackers against the prone dodging creature would not have advantage or disadvantage, while ranged attackers would have disadvantage.

There are additional rules on "Being Prone" (PHB, pp. 190-191), but it is mostly describing how to move while prone, and does not forbid the use of any actions, including the Dodge action.

(Incidentally, I'm imagining it would look like Samwise trying to dodge Shelob's stinger from the Lord of the Rings: The Return of the King film.)