Mage Hand's description reads:
You can use your action to control the hand. You can use the hand to 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.
Fast Hands:
Starting at 3rd level, you can use the bonus action granted by your Cunning Action to make a Dexterity (Sleight of Hand) check, use your thieves’ tools to disarm a trap or open a lock, or take the Use an Object action.
Presumably, this newfound dexterity is proper to the Rogue (i.e., the physical prowess of the PC), whereas Mage Hand is based on mental effort and the limitations of the spell. Hence, without a houserule, I don't think you can control Mage Hand using a bonus action.
As othniel2005 said, you would have to pick the Arcane Trickster archetype for that.
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.
Best Answer
Your assumption is correct. Unlike, for example, Spiritual Weapon, Mage Hand doesn't say anything about a free use on the turn you cast it, so it requires your action to cast it followed by your action on a subsequent turn to use it.
Minor caveat: An Arcane Trickster can control a Mage Hand with a bonus action, so they could create with their action then use it with their bonus action. Also, a Fighter could use Action Surge to gain an extra action and use it on the same turn.