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.
Mage hand can pour out a vial of acid.
Pouring a vial of acid in this way will (generally) not damage an enemy.
It's exactly as the rules you quoted say. You can pour out the contents of a vial. Per the description of Acid, simply pouring isn't enough to do someone--who likely doesn't want acid poured on them--damage.
If you start pouring acid on one who doesn't want to be acidified, they can move, interpose an object, or otherwise evade the damage. In order to damage someone with acid you've got to splash it onto them, an activity that's defined as a ranged attack. And Mage Hand, per its description, can't effect damage on someone trying to not be damaged: it can't attack.
Could one pour the vial over an unsuspecting enemy, or one incapacitated or restrained, thereby causing them damage? I contend it follows the same rules: if you would need to make an attack roll, Mage Hand can't do it. Whether those situations require an attack roll--and therefore are attacks which Mage Hand cannot effect--is a ruling to be made by the GM.
Best Answer
For a Wizard? No.
The basic text of Mage Hand says it can manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial. It cannot attack, activate magic items, or carry more than 10 pounds.
Disarming Attack1 is a special ability available to Battle Master Fighters or characters who take Martial Adept (PHB p168). Either way, it requires hitting the opponent with a weapon attack. Mage Hand cannot make attacks and is also not a weapon, so that's out.
Shove (PHB p194) is also an attack, so no.
For an Arcane Trickster Rogue? Some of them.
Stealing other items requires looking at the things an Arcane Trickster Rogue (PHB p98) adds to Mage Hand.
If the specific rules2 for Mage Hand Legerdemain have to call out the ability to stow and retrieve objects from other people, then clearly the intent of the general case is that Mage Hand alone cannot manipulate objects in another creature's possession.
Mage Hand Legerdemain only allows for a Sleight of Hand (Dexterity) check to do the specific things it lists. It does not establish a general case for making ability checks with Mage Hand. An Arcane Trickster has to reach level 13 before he can even use his Mage Hand to be annoying, as a distraction via Versatile Trickster (PHB p98).
Why isn't it a basic "object interaction"?
Can Mage Hand be used on an unattended object? Absolutely. Can it be used on an object in the possession of another creature? That's pick-pocketing, something that requires a Sleight of Hand (Dexterity) check. If taking things off a person is a basic object interaction, any character could walk up to another and take their arrows, and nobody could stop them.
1DMG Optional Disarm? Still an attack, still requires a weapon, still not possible.
2It may be worth noting that Mage Hand Legerdemain does not have the caveat of "open container" that Mage Hand does, another advantage that an Arcane Trickster has over a Wizard.