No, the first two points don't hold for any caster with Mage Hand.
Pickpocketing isn't just taking something from a pocket; it's a subtle skill which requires doing so unnoticed, after all. This is more than just "doing it while the victim's back is turned" - the human body registers slight touches and subtle sensations, like the weight of an object. A pickpocket knows how to beat these senses - by touching the victim in other places to confuse the senses, for example.
But the Mage Hand is implied to be clumsy, with limited dexterity - not conducive to stealth.
(Note that nothing in the Mage Hand description says that the hand has significant dexterity - it can "open an unlocked door", for example, or "pour the contents out of a vial", but apparently not manipulate lockpicks or disarm traps, which requires more precision. It's limited almost exactly to simple tasks that you don't have to practice.)
So it's not that the Mage Hand eliminates the normal skill check as such - instead, the rules as written do not contain a "normal" skill check for picking a pocket with a Mage Hand. (A standard thief can't pick locks with a Mage Hand either, even if they somehow have one.)
Most people can undo the buckle on a bag, or shoe. But can you do so so swiftly and subtly that the holder of the bag (or wearer of the shoe) doesn't even notice? Try it. It's not as easy as it sounds. It needs significantly more manual dexterity than just being able to undo a buckle - it's more like being able to play a piano.
Under the old-school skill system of 3e or 4e, the correct way to handle this would have been with a skill roll and an extremely hard DC - as GM, personally I'd have set the DC 10 or even 20 points higher than the usual for that kind of pickpocketing. The old skill system would then have allowed top-level characters to pull it off anyway.
But 5e discourages this "everything is technically possible with the right roll" approach, in favour of limiting skills to "actions anyone could attempt". "Pick a pocket with a magical force" isn't something anyone could attempt, and there's no obvious RAW reason why being able to pick pockets by the normal means would help you use a Mage Hand to do so.
(And it is typically next to impossible to pickpocket successfully with a fully visible, somewhat clumsy, disembodied hand. So even a disadvantage is not enough penalty - it should just be impossible.)
Legerdemain clearly gives the hand more dexterity, not just invisibility. This allows for more complex tasks.
Short answer
If the standard hand doesn't have enough manual dexterity to pick locks or disarm traps, it doesn't have enough for the equally tricky task of picking pockets.
Probably not. Tavern Brawler requires you to "hit the creature" with a weapon attack. While the Disarm rule involves making an attack, it's not clear that that attack involves hitting a creature. Normally, hitting a creature requires beating its AC:
When you make an attack, your attack roll determines
whether the attack hits or misses. To make an attack roll,
roll a d20 and add the appropriate modifiers. If the total
of the roll plus modifiers equals or exceeds the target’s
Armor Class (AC), the attack hits. The AC of a character
is determined at character creation, whereas the AC of a
monster is in its stat block.
Since the attack you make to Disarm isn't against a creature's AC, it probably doesn't involve hitting a creature, and therefore can't trigger Tavern Brawler.
Best Answer
You can pick it up.
The question you quote is correct that your enemy can just pick the item back up, but they have to do it on their turn. You can pick it up on the same turn you do the disarm, though (PHB 190):
In the table on the same page, labeled "Interacting with Objects Around You," one of the options is
Your disarm is part of your action, so you can pick it up alongside this action. Now that you've stolen the disarmed item, the enemy has to disarm you to get it back.
You can stash it.
It's a bit ambiguous whether or not you can stow the item in the same round without expending an action on it. The same page in the PHB states,
Since picking up the item and stowing it is interacting with only one item, one could easily argue that the entire thing falls under a single item interaction. If that is the case, then your enemy would have to do something else to get the item back, as you're no longer holding it. Alternatively, since it's two different actions you're taking with the same item, a DM could say that it costs an action. It's ultimately up to the DM anyway:
Personally, I would rule differently depending on the item. If it was a small orb or something, I would let a character put it away without a second item interaction, but if it was a warhammer, I would require an action.
Your teammates can damage it.
If you have a teammate that goes before them, they could also pick up the item or attack it or something before the enemy can retrieve it.
For example, Fire Storm states,
and therefore would damage or destroy the object once it's dropped. Similar spells like Fireball have similar wording. Indeed, if someone were to cast Fire Storm on the square in which your enemy was standing, the disarmed object would catch on fire as soon as it is no longer carried.
For completeness, the PHB does allow targeting objects with attack rolls. On pg. 194, in the "Make an Attack" section, the first option is: