While at first it doesn't seem an efficient trap given the fact that animated armor will need to spend part of its time establishing control. It does have a nasty aspect that the PC will be inside forcing the rest of the party to come up with a innovated way of stopping the armor without hurting (much) or killing the character inside.

Should the Animated Armor's control over the PC be implemented via the grapple mechanic? Is that technically feasible? If so, I feel like either the PC should get disadvantage or the Armor should get advantage. Is there a significant difference statistically in the outcome depending on which one of those I choose?

Basically yes, at its core should involve a contest between the two opponent's strength so the grapple use of the strength based skill Athletics is a good choice. However I would rule that can't use Acrobatics once the PC is inside the armor. Only Athletic checks are allowed.

The same with the escape attempt, once inside the only way to regain control is with a Athletic (Str) check.

If the grapple mechanic is used to implement this tactic, would it be unfair to grant the Armor the "Damage Transfer" property of the Rug of Smothering? I feel like attacks directed at the armor should also harm the wearer, but effectively granting the armor complete damage resistance (allowing it to only take half damage from virtually all attacks) seems pretty brutal.

The Damage Transfer ability of the Rug of Smothering means that it takes longer to kill the creature. The target inside the Rug will take the same amount of damage needed to kill the Rug itself.

Now that I think about it the Damage Transfer ability was probably granted to the Rug in order to prolong the danger from the Rug's Smother attack. In light of this my recommendation is that for the Animated Armor trap, damage be equally allocated between armor and the character inside. However remember the character inside doesn't have the poison or psychic immunity the armor has.

While the Armor is controlling a PC, should the "combined creature" (combination of PC and Armor) attack using the PC's attributes and attacks or the Armor's attributes and attacks? For example, take a fighter who wields a Pike who has been taken over by a suit of Animated Armor. Can the Armor make attacks with the Pike using the fighter's Strength, or is the Armor forever fated to only be able to use it's default Slam attack with its own Strength score?

No that doesn't make logical sense. The way you describe it the Animated Armor is acting like an sci-fi exoskeleton with a will of its own.

Similar to #3, should the combined creature use the PC's AC or the Armor's AC? For example, assume a fighter wielding a shield and a Ring of Protection +1 straps on the Animated Armor and loses his grapple check, his total AC would be 21 (base 18 from Full Plate Mail, +2 from the shield, +1 from the Ring of Protection). Should attacks against the PC+Armor be targeted against an AC of 18 (for just the Armor) or 21 (for the PC)?

Basically yes except that the character inside would not get a dex bonus to their AC or use of their shield.. If they still have a higher AC, I would rule that the only the Armor suffers damage if the attack is equal to or greater than the Animated Armor AC but less than the PC dex-less AC. If the PC had a shield it would be used by the Animated Armor making it a AC 20 but not the PC.

Final Thoughts

I would rule it like this

1. The PC puts on the Armor
2. The armor comes to life initiates a grapple check.
3. If the PC succeeds, he yells help and we go back to step 2. The PC will get his normal turn.
4. If the PC fails, he goes oh crap and still yells help, and now has the restrained condition similar to the result of the Rug of Smothering's smother attack.
5. The armor uses it interaction to pick up the PC's weapon. If the PC has a shield then it will spend its action to pick up both weapon and shield. The armor then moves and does it action (if has one)
6. PC can attempt an escape with Athletics only.
7. The armor can takes it turns as long as the PC doesn't succeed on an Athletic check. The PC does then the Armor will return to fighting for control.
8. If the PC makes a number of successful Athletics, your call, then he has managed to rip the armor. If the party aids him then he makes the check with advantage. I recommend at least two in a roll, no more than four in a row.

This would refer to armor that was primarily metal. As in, you would look at it and say "that armor is made of metal." As opposed to "That is cloth armor" or "that is leather armor." This is supported by a quotation from the Sage Advice column, where it is talking directly about the Druidic Taboo against wearing metal armor.

The idea is that druids prefer to be protected by animal skins, wood, and other natural materials that aren’t the worked metal that is associated with civilization. Druids don’t lack the ability to wear metal armor. They choose not to wear it.

[...]

A druid typically wears leather, studded leather, or hide armor, and if a druid comes across scale mail made of a material other than metal, the druid might wear it.

http://dnd.wizards.com/articles/features/rules-answers-march-2016

Here, we can see that the metal fittings that would normally be added to studded leather do not count as 'Metal Armor.' And, strictly speaking, you can make Leather Armor with fittings other than metal rivets. I am actually a hobbyist Leatherworker and I do know how to make armor. Ideally, you use metal rivets to hold the pieces together...but it's entirely possible to use stitching, bone, or other hard materials in place of the metal.

For a simple test, just read the description of the armor in the PHB. Every description emphasizes the primary material that the armor is made from. For example...

Studded Leather. Made from tough but flexible leather,

and

Chain Shirt. Made of interlocking metal rings[...]

and as a final interesting point...

Ring Mail. This armor is leather armor with heavy rings sewn into it [...]

In short, while most of this will be DM-fiat, and it's quite possible to have Dragonscale Scale Mail or a Breastplate made from the carapace of some insectoid creature...as a general rule of thumb, if the description of the armor in the PHB explicitly mentions metal, then you can likely assume that piece of armor counts as 'metal' for the purposes of Druids, Spells, etc.