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.

The Rules Don't Say

As nvoigt pointed out, the PHB describes chain mail and plate mail as complete items. They are either on, or they're off. There are rules for how long it takes to put them on and take them off, but there is no description for your AC while wearing just the padding.

Furthermore, the padding under plate and chainmail isn't padded armor. Padded armor is a different thing. When you buy plate mail you don't get plate mail armor that has padded armor inside, you get plate mail armor that has padding inside.

You Could Try It

Since there are no rules that describe this situation, you are down to the most general rule of all. At the beginning of the Players Handbook, it says:

1.The DM describes the environment. ... 2.The players describe what they want to do. ... 3.The DM narrates the results of the adventurers' actions. (PHB p.6, "How to Play")

So, your character could just try it and see what happens. "Try it" could mean a lot of things. "My character takes off all his armor but the padding. What's his AC?". "I'm looking at the swamp we're about to cross. I'm an experienced fighter, I've been wearing armor for years, I've been trained by experts. I know if I fall in wearing full plate, I'll probably drown before I can get out of it. What does my training tell me about whether I can wear just the padding in order to mitigate the drowning risk but still retain some protection?" Or perhaps your armor wearer might consult armor experts.

You've Solved My Drowning-in-Armor Puzzle!

As a GM, I view the water-armor situation as a challenge or problem for the PCs. Which do you do, take off the armor and carry it thereby lowering your AC, or wear it and risk drowning?

Only it isn't an either-or, which is the beauty of RPGs. There are other solutions to the problem. Perhaps the water is such and the PC's strength is such that the PC can just walk out, perhaps with a strength or constitution check. Perhaps the PCs have access to magic that allows them to breathe water. Maybe they can bypass the swamp. The players might (and hopefully will!) have innovative solutions I haven't thought of.

If the PC said, "I take off most of my armor, and wear only the padding", or "only the helmet, gauntlets, greaves and leg guards" and subsequently fell into the water I think the time to remove the armor and thus the risk of drowning would be less, while the chances of swimming in part of the armor would be more. And if the PC were attacked by Rodents of Unusual Size while crossing the swamp, I'd definitely have to allow that the PC's base AC in padding or other bits and pieces was greater than without armor. Wearing only part of the armor might be a reasonable solution to the but-I-might-drown problem. Although wearing only the padding might be really bad for it, given briars and such.

Given that I don't want to play "Armor & Anoxia" I'd be reluctant to come up with a codified houserule that says you can wear half the armor for half the AC and half the risk of drowning, but perhaps that would be the solution. I'd definitely want to keep things liquid, and not weigh the game down with heavy armor questions, but I'd definitely want to pad it enough to keep things moving forward, while at the same time rewarding innovative role-play and problem-solving.