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 Legerdemain
Starting at 3rd level, when you cast mage hand, you can make the spectral hand invisible, and you can perform the following additional tasks with it:
• You can stow one object the hand is holding in a container w orn 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.
You can perform one of these tasks without being noticed by a creature if you succeed on a Dexterity (Sleight of Hand) check contested by the creature’s Wisdom (Perception) check.
In addition, you can use the bonus action granted by your Cunning Action to control the hand.
"But how does the rogue know its limits?"
The mage hand spell and the Mage Hand Ledgerdemain description give the general idea. Past that, the player describes what he or she wants to do, and the the DM narrates the results.
"Do they mentally KNOW it's location?"
That is not specified in the rules, so up the the DM. I rule it that the rogue just knows where it is.
Another interpretation might be the rogue can "see" an outline, like an invisible item in a cartoon is sometimes depicted.
"Can they feel through it?"
There's nothing in the rules that explicitly says they can feel through it, but the rules imply there is enough sensory feedback to accomplish the tasks listed in the description. I allow the rogue enough sensory feedback to accomplish the task, but not enough sensory feedback to accomplish something very different from the tasks listed.
"If so, does it extend to all senses of touch like temperature, pressure, texture?"
Same for these.
"I know it can't pass through objects but could they send it through a window or chimney and open a door locked on the other side?"
Maybe, depending on the situation and the DM's interpretation. It would be reasonable for the DM to give a significant additional difficulty rating or to make it several separate tasks and checks.
"Pick someones pocket from the opposite side of a busy room?"
Yes. Or rather, the rogue can try. Increased DC or disadvantage might apply, especially if vision is obscured. Or maybe it is several different checks. Maybe one to even find the pocket. Maybe the rogue gets the wrong item. It shouldn't be really easy to pick a pocket that you can't even see, but maybe not impossible. The risk of the target catching on that something is going on should be significant.
"Can they disarm people since a weapon isn't in a container?"
No, that's an attack.
"Steal a ring of keys?"
As in pick it up and carry it? Yes. As in remove them from a container? Yes. Those are all within the description.
In general, it is easy to interpret MHL very broadly, and there is nothing wrong with that. The DM has to decide where the limit is, based on what sort of campaign you and your players want. The rule of fun is always a good one. But it is also worthwhile to keep balance in mind. The Arcane Trickster has lots of other features besides MHL.
Imagine this scenario. The rogue is in a jail cell. The rogue knows the guard is sleeping around the corner and saw that the keys are on the guard's belt and earlier saw where the chair was and what the belt and keys looked like. "I use my MHL to get the keys!" "You can't even see the guard, you use MHL to feel around. Roll a perception check." "16!" "It takes you quite a while to feel where the keys are, but you also hear the guard moving in his sleep. Maybe you're tickling him. You try to remove the keys. Make a sleight of hand roll with disadvantage." "Rats, 12!" "That won't do it. You can't get the keys off. You've used up 8 rounds, you've got two tries left." "Arrgh! 8!" "Not even close. Suddenly the guard isn't making any more noise." "Did he wake up?" "How would you know?" "Try again or not, the spell's about to lapse." "21!" "Seriously, that was with disadvantage?" "Oh, right. Oh baby, yes! 20!" "You feel the keys lift off the guard's belt, and they come floating around the corner. They're within two feet of the jail cell, and the spell ends, and the keys fall to the floor. It sounds really loud. The guard yells what the!" "I reach out and grab the keys and hide them!"
To me this makes it fun. The guard is sleeping, but the rogue can't see the keys. The MHL isn't an auto-win, but the rogue gets to use it. PCs are hard to keep in jail, after all.
Best Answer
The hand is sneaky. The process of casting is not
The Mage Hand Legerdemain feature description says:
So you need to cast the Mage Hand normally. The spell description says:
That means the spell has Verbal ("the chanting of mystic words") and Somatic ("a forceful gesticulation or an intricate set of gestures") components. You can't perform them unnoticed during a conversation. If the other person identifies the spell itself, depends on the person.
Related questions:
It remains unclear though, how obvious is a process of controlling the hand. The spell description only says that
So when you control it, you must expend an action (or bonus action, for the Legerdemain version). Probably you have to perform special gestures, sort of. The exact details depend on the DM.