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.
Yes, based on the "fine control" statement
If we look at the examples that are given for "fine control," they could potentially involve holding things in place. For example, pouring out the contents of a vial would require holding the vial still (or still enough), and manipulating a simple tool could potentially require pushing the tool against a surface, holding it still. Additionally, there are no stated limits on that fine control, so there aren't any. After all, this is magic, not muscle power; our only limits are the ones explicitly delineated in the spell and the fundamental system restrictions.
Additionally, requiring that an object be in motion doesn't make sense. If you're trying to remove an item from a creature's grip, what happens if the two opposing forces cancel out and the item doesn't move?
Replicating Levitation is fine
You want to differentiate between levitation and telekinesis, but levitation is really just a subset of telekinesis. Note that telekinesis is a 5th level spell, but levitation is only 2nd level. Thus, it seems balanced to allow your player to replicate the effects of a lower level spell using a higher level spell.
There are many high level spells that could be used to duplicate lower level spells. Wish is the archetypical example, but the various resurrection spells and the "mass" spells also fall into this category.
Your player might be confused about the ring, and he might not use it to its full potential. Still, it makes more sense to make sure that he knows the full extent of telekinesis, rather than banning his preferred usage of that power.
Best Answer
No. The authors are careful about denoting when spells can be used for an attack.
There is nothing in the game rules more important than the action economy. Over and over again, the authors painstakingly indicate how spells and powers fit into the structure of "attack actions", "bonus actions", "movement", "dash", and so on. Given that there is no indication that telekinesis allows an "attack action" with the object, you can be assured that the authors meant there not to be.
There are times when the rules should be set aside in favor of "real" physics or dramatic license. However, the DM should set the bar quite high for this. The first question you should ask yourself should be, "Is my player asking for something that people like Jeremy Crawford and Mike Mearls didn't think of?" Rest assured, they thought of telekinesing a dagger across the room or a "dancing" sword. In fact, they designed spells that do as much. It is implausible that they did not consider the possibility and then failed to address it in the description.