No, it does not.

The Dispel Magic description says:

Choose one object, creature, or magic effect within range. Any spell of 3rd level or lower on the target ends.

And Animate Dead has a duration of:

Duration: Instantaneous

So there is no spell effect to dispel as Animate Dead has a duration of instantaneous, so as Dispel Magic only works on a spell it has nothing to work on and does nothing. Also clearly stated here stated in this Sage Advice:

Can you use dispel magic on the creations of a spell like animate dead or affect those creations with antimagic field? Whenever you wonder whether a spell’s effects can be dispelled or suspended, you need to answer one question: is the spell’s duration instantaneous? If the answer is yes, there is nothing to dispel or suspend.

The control of the undead granted by the spell is a consequence of the spell being cast, not an on-going magical spell, in the same sense as the fires started by a fireball are an on-going non-magical effect. The zombies are strictly loyal to the caster for a period of time, bonded to them like ducklings are to their mother, and in the same way the fires and damage from a fireball last for a period of time and cannot be dispelled, neither can this bond.

Animate dead wouldn't work; the creature's actual type is unchanged.

The animate dead spell must be cast on "a pile of bones or a corpse of a Medium or Small humanoid within range", as you quoted - but the feign death spell doesn't turn a creature into a corpse (or a pile of bones).

Feign death says "the target appears dead to all outward inspection and to spells used to determine the target’s status." However, the animate dead spell is not a spell used to determine the target's status; it is simply a spell that only works on dead creatures. In other words, it is a prerequisite/requirement of the spell - not a function of the spell.

Because feign death doesn't change the actual status of the target, the target of feign death is not a valid target for the animate dead spell. One way to handle this situation is using the optional rule on invalid spell targets from Xanathar's Guide to Everything (p. 85):

If you cast a spell on someone or something that can’t be affected by the spell, nothing happens to that target, but if you used a spell slot to cast the spell, the slot is still expended. If the spell normally has no effect on a target that succeeds on a saving throw, the invalid target appears to have succeeded on its saving throw, even though it didn’t attempt one (giving no hint that the creature is in fact an invalid target). Otherwise, you perceive that the spell did nothing to the target.

Following this rule, nothing would happen to the target after animate dead was cast; the spell slot would be expended, but animate dead would fail. However, no indication is given as to why the spell failed; as Miniman pointed out in a comment on this answer, another reason it could fail is that it's under the effect of a gentle repose spell, for example.