First some background. When you attempt to touch someone with a spell, you are considered armed:
Touching an opponent with a touch spell is considered to be an armed attack and therefore does not provoke attacks of opportunity.
You can also discharge the spell with an unarmed strike, but this does provoke an AoO:
you may make a normal unarmed attack (or an attack with a natural weapon) while holding a charge. In this case, you aren't considered armed and you provoke attacks of opportunity as normal for the attack. If your unarmed attack or natural weapon attack normally doesn't provoke attacks of opportunity, neither does this attack.
So if you have some way of avoiding the normal AoO an unarmed strike provokes, it will let you avoid it here.
"With this, would striking someone with (spiked) gauntlets be considered "touching" him?"
Sadly, spiked gauntlets are not such a way; they are basically a weapon (the spikes) attached to your gauntlet. Contrast that with regular gauntlets, which simply modify your unarmed strike, and are classified as such in the list of weapons.
That said, it's a reasonable thing to bring up to your DM as a house rule. Your other options are to take the feat unarmed strike, or to somehow gain natural attacks.
"When I learn, as a druid, to cast spells while being in wild shape, can I attack an enemy using Produce Flame with a natural attack?"
Yes, this is covered by the rules I've quoted above. A natural weapon doesn't normally provoke, so neither would using it in conjunction with spells like produce flame.
There is no Opportunity Attack
The Polearm Master Feat is clear that the reactionary attack is an Opportunity Attack, and Disengage says that your movement doesn't cause Opportunity Attacks.
In your example, moving into a Polearm Master's reach while Disengaging doesn't trigger the Feat's Opportunity Attack. Think of it as sort of batting away the guy's glaive before moving in.
Disengage should work this way, otherwise a Polearm Master can always attempt to Opportunity Attack against an approaching foe.
Best Answer
You would not provoke any Opportunity Attacks using this method (and neither would the hand)
The PHB tells us when an opportunity attack is provoked:
So you are correct in saying that the unconscious carried creature would not provoke any Opportunity Attacks from enemies whose reach it leaves because it is not using movement, actions, or reactions to move. So enemies can't lay a finger on you in this way.
Thus, this plan works as you say.
Side note: RAW Bigby's Hand cannot provoke OAs
I realize this is a bit nonintuitive, but as written there is no way for the hand to provoke OAs because it is not a creature. Per the description of the Bigby's hand spell (or arcane hand, as it is listed in the SRD):
And only creatures provoke OAs, per the description of opportunity attacks:
Even if you houseruled that the hand could be handled as a creature, it still would not provoke OAs because of the rule we cited earlier. Since the hand is not moving using its action, reaction, or movement to relocate because the caster is using their bonus action to control and move it.
It is your bonus action that you spend to move the hand. The hand does not even have any actions or movement of its own that it can use.
So, unintuitive as it may seem, the hand cannot provoke OAs because it is an object and because it doesn't use its own actions or movement to move.1
So, yes the technique works and avoids OAs against the carried creature and the hand and is a clever and handy strategy to use in cases like the one in question.
1 - Note: Even if you houseruled that the hand provokes OAs (despite all the rules reasons that it should not), the attacks would have to be directed at the hand itself and not at the carried creature. There is no way by default to redirect an OA at another target that did not provoke an attack.