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.
This is situational.
I don't do two separate rolls, when it's a standard pickpocket or reverse pickpocket situation, IF the initial Sleight of hand is a success, but that's a DM call, as it is written.
So for instance, you want to reverse pickpocket a letter into someone's pocket. First, I have you roll to see if they notice--Dexterity (Sleight of Hand) check contested by the creature’s Wisdom (Perception) check.. If they don't notice and the sleight of hand is a success, I assume it's a success as far as it being carried out. (This makes sense to me because if it goes smoothly it won't be noticed.)
But if they DO notice, I THEN have the player roll to see if they got the job done anyway. So, you're picking their pocket, they notice, but you manage to get the wallet just as they notice. That would be Dexterity (Sleight of Hand) check, difficulty determined by DM, not contested by the victim. The victim might later be able to get it back or something but that's a whole different set of rolls.
If you're reverse pickpocketing, say, a damning letter onto their person, they can notice, but it might get in a pocket anyway. Of course, afterwards they'll take it out. But they might still have it in their hands at the right time. If it's some kind of magical incendiary they may have time to toss it away before it goes off. (They would roll for that themselves separately).
The exception to this rule, is if I am building suspense and it's particularly difficult--like line of sight is coming in and out because of a crowd of something. In those cases, the subject might not notice, but it's still possible to fail, and for the player to try again. In this case, I reverse the order of the rolls--first they roll for successfully pick pocketing, and then I have them roll the sleight of hand. It builds suspense, because the player can fail to pickpocket, but the subject still might be unaware, and the player can try multiple times.
With picking a lock at a distance, going unnoticed is totally separate from a success or not, depending on how close the noticer is to the lock.
The distance of the person noticing from the lock, determines whether them noticing would have any impact on success. So, if the person noticing the lockpicking is less than 5 feet away, I have them roll for the sleight of hand, and that might stop the attempt--basically, I handle it like pickpocketing.
But, if the guards are more than 5 feet away, I do both rolls at the same time, the sleight of hand, and the roll to pick the lock. Both are done at the same time, and while the person noticing might react, they are unlikely to prevent the success or failure of the person trying. Generally I give the player two dice, and assign different colors to the different things, and they roll them simultaneously. That's just DM flair.
Ultimately, it's up to the DM to decide what's fair, but this is how I handle these. Technically, and I do think you want to be technical here, there are supposed to be two rolls. One to see if you are noticed, and one to see if you are successful. I just handle it situationally, the better not to waste time on things, or the better to build suspense.
Best Answer
Yes.
Anyone can attempt to use Sleight of Hand to steal something mid-combat. For most PCs, this will require their action to make this attempt. Others may be able to use their bonus action.
As for whether or not the creature uses its passive Perception or a contested check would largely fall to the DM's discretion. Personally, I would look to the description in the rogue class for this sort of action since two of the roguish archetypes in the PHB specifically call out Sleight of Hand.
First, the Thief archetype on PHB p.97:
and on PHB p.98 under the Arcane Trickster archetype (emphasis mine):
In the case of Fast Hands, it is not specified and the choice of passive/contested would therefore be left up to the DM. For Mage Hand Legerdemain, it specifically states that it would be a contested check.
As a DM, I would first consider whether the creature was aware of your presence. In a case where you have not been seen (you were hiding before combat began), I would probably use the passive perception. However, if the creature is aware you are out there somewhere (you hid after it initially saw you) then it makes sense to me to use a contested check because the creature would be actively looking for threats at that point. If you weren't hidden at all and attempted to steal an item while engaged with the creature, I would probably also apply disadvantage if you couldn't give me a reasonable explanation of how you might do it without them noticing (see examples below). And of course, there are some items that I may rule were impossible to steal at that point in time.
Example 1:
The PC moves as if to grab the jailer, but it is really a distraction so he can palm the key sticking out from the jailer's vest pocket.
Example 2:
A rogue attacks with his main hand. At the end of his attack action, while the creature is still distracted by the attack weapon, his off hand darts in during his bonus action to steal a potion.
Example 3:
The barbarian attempts to steal a dagger from their opponent's belt sheath without any distraction at all.
In examples 1 and 2, I would allow them to roll contested checks. Example 3 would be rolled at disadvantage.
One last note, I would apply these same principles to any Sleight of Hand attempt used outside combat as well.