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.
You do not need to be able to see the Mage Hand to be able to use it, otherwise ATs wouldn't be able to make the hand go invisible. However, working around corners (or otherwise out of sight) would effectively impart the blinded condition to any action you were going for. As such, you wouldn't be able to interact with a target except by guessing which location it was in (unless you knew the target's location, and it was static (i.e. an unmoving object (keys hanging on an post) or a guard that's asleep in a chair)).
Nothing about the spell itself implies any sensory input gained from the hand, so unless you can see what's going on to direct it, it's going to be pretty difficult to use. As a caveat to that, however: ATs can make the hand go invisible... therefore, they have to be able to at least "feel" where it's at.
If the door was simply a one-way door that didn't actually require a key, I'd say (unless you rule that the handle requires more than 10 lbs of pressure to activate) that it would work if she could get the hand into the room.
If she was suggesting picking the lock, I would say that's not possible because she couldn't get her mage hand or lockpicks to the other side of the door... unless she casts/pushes them through the keyhole/under the door (your ruling on whether the keyhole goes all the way through (in which case, she could pick it from her end)).
RAW: With an invisible mage hand, you can do the following (AT, Mage Hand Legerdemain, PHB p. 98)
- You can stow one object the hand is holding in a container worn or
carried by another creature.
- You can retrieve an object in a container worn or carried by
another creature.
- You can use thieves' tools to pick locks and disarm traps at range.
In the spell text, it doesn't put any restriction on "any object not worn or carried," so it could definitely be used to take a crossbow bolt, or even pull the trigger prematurely.
RAI: You couldn't possibly see what you're doing with a fine lock or trap at range, so being blinded isn't an issue, because you can "feel" it through the hand. And you don't need to be able to see the hand. If she could get the tools, she could open the door from the other side.
As far as harassing people... that's pretty broad. You could apply 10 pounds of effort in quite a few fashions. Steal a handful of their arrows (and hold them up in the air (no more than 30 feet away from you)), undo their belt, tie their shoes together (or any other myriad wardrobe malfunctions), pull their hair or flick their ears(no damage), put a thumbs-up in their chair as they're about to sit down, use scissors to snip a cross/bow string (cruel, if it's a magic item - maybe only against mundane strings)... the list goes on.
Best Answer
Unclear, though game designers seem (unofficially) ok with it
The list of things that Mage Hand can do is quite short, but the terms are quite broad. As quoted in your answer (and from PHB, p. 256):
It could be argued that you could use a Healer's Kit by "manipulating" it: after all, the first definition of "manipulate" via Oxford Languages (the dictionary service Google draws from at the time of this answer) is:
It certainly sounds like this would give you the ability to skillfully use the Healer's Kit with Mage Hand. But there are a few impediments to this idea.
Can you use the hand to do something that normally takes an Action?
Since you needed to "use your action to control the hand," it's reasonable to ask whether the "use" of the hand could include something that normally requires an Action: after all, you have no Actions left (barring the use of a feature that gives you a second one, like Action Surge). While this is an open question, we have some guidance in the list of prohibited uses, specifically the rule that (ibid):
The interesting thing about this prohibition is that attacking normally also takes an Action (typically, the Attack Action). So this text implies that there might be some forms of "manipulating an object" that (normally) cost an Action that mage hand can perform: otherwise, they wouldn't need to point out the Action-costing things that it can't do. This is known as an "exception that proves the rule."
It's not quite that clear cut, though. For example, the fact that some uses of mage hand explicitly allowed to an Arcane Trickster (e.g. "You can use thieves' tools to pick locks and disarm traps at range", PHB p. 98) could be done by "manipulating an object," but you clearly can't do them with regular Mage Hand since they are "additional tasks" (ibid) that an Arcane Trickster can do and other people casting Mage Hand cannot. So there are definitely Actions the hand cannot do which aren't listed in its spell description.
All in all, there is room to interpret the spell in either direction. But there is enough wiggle room that it is possible the hand could perform certain activities that normally would require a full Action to do, even though it already takes an Action to control the hand.
Does the Healer's Kit count as "an object"?
On the face of it, definitely not. The text states (PHB, p. 151):
So the kit is more than one object: it is a 3 pound collection of objects in a leather pouch. However, it's worth noting that when you are stabilizing a creature with the kit, we're not sure how many of these objects you actually use! Perhaps only one of the objects in the kit is needed for particular types of injuries (e.g. a salve for slashing damage, a splint for bludgeoning damage), or perhaps you need to use multiple ones for each type.
Now, medically, it would be entirely unrealistic to suggest that most treatments for serious wounds could be achieved with a single isolated object (I only have first aid training, but still this is easily apparent). Even bandages are actually several layered items, often gauze followed by a bandage to hold it on, sometimes preceded by salve. MivaScott points this out clearly and convincingly in their excellent answer. However, the application of actual medical logic is complicated here, because no realistic assessment of injuries is going to match with this game's rules for their treatment (e.g. no realistic injury system would have a person be entirely unimpaired in terms of their ability to perform tasks after one hour of unconsciousness, when they were previously inches from death).
Also, unlike any real world medical interventions, a Healer's Kit is (normally infallible at stabilizing a dying patient: it requires no skill Check to succeed, or proficiency to use, but simply always works (as long as the user expends one of its ten "uses"). Thus, we have real world medical materials (splints, salve, etc.) being applied in an unrealistic way: it's unclear exactly how this is done, and thus how many "objects" are involved. At best, we need to see this as another case of ambiguity.
So if the wording is all ambiguous, whom can we turn to? (The people who wrote the words)
In times of ambiguity, it can be useful to turn to designer commentary to clarify the rules. And there is designer commentary on this question, but it is the most unofficial and qualified kind. Specifically, the following exchange occurred on twitter with Jeremy Crawford (lead designer of the Player's Handbook):
Now, first of all, tweets by game designers are unofficial guidance in general, unless they are collected into the Sage Advice Compendium (which this is not). And second of all, by saying "I'd allow," Crawford is indicating what he would permit as a DM, which can go beyond rules interpretations (and via "rule 0" can even contradict the Rules as Written).1 Specifically, saying "I'd allow" means he is making a judgement call which is not demanded by the rules. That could mean that the rules are silent or unclear on the issue (which certainly seems to be the case to me), or that the rules require some other conclusion but he is overriding them as DM. In the absence of further commentary2, we have no way of knowing which way he meant the comment.
However, in the absence of other clear guidance, this exchange could be helpful. It shows that it may be possible to use the kit with Mage Hand, but there is enough ambiguity that such a decision is ultimately up to a DM.
And honestly, that's the best answer I can give to this question. The rules are sufficiently ambiguous that a DM will need to make their own call. But if, for whatever reason, you'd like to make the same call as the person who wrote most of the game (and I stress: you do not need to do that as a DM if you do not want to, and that decision was made by him as a DM, not as a game designer), then you should allow Mage Hand to use a Healer's kit to stabilize at range.
1.) It's also worth noting that the caveat written into his response (that mage hand can use a healer's kit "it's not holding") points to a possible high cost for this tactic in terms of the action economy. The text of the spell indicates it might take an action to open the kit ("open a... container"), another action to remove the object needed to treat the wound ("stow or retrieve an item from an open container") and a third action to actually use that item (" manipulate an object"). It also takes an action to even cast the spell, and the consensus (which I'd probably rule differently as a DM but is accurate via RAW) is that the Action taken to cast Mage Hand is separate from the Action required to control it. Thus, depending on your DM's assessment of how a Healer's Kit is "used," it could take three or four rounds for someone to "use" the kit with Mage Hand (which is a long time for a creature to be unstable and dying).
2.) Note that a twitter user did reply to the aforementioned tweets by Crawford, asking "Is this ruling on Mage hand and a Healer's Kit RAI or a house rule for your personal game?" Crawford has not replied to that follow-up question.