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.
Clever, but unlikely
In order for Catapult to work, it requires:
one object weighing 1 to 5 pounds within range that isn’t being worn or carried.
This is the requirement that Delayed Blast Fireball must meet. The question is: Does it?
Delayed Blast Fireball isn't explicit in the definition, but it does state:
A beam of yellow light flashes from your pointing finger, then condenses to linger at a chosen point within range as a glowing bead for the duration.
How much does a bead of light weigh?
In order for that bead to qualify for Catapult, it must be at least one pound and less than five pounds. As Dale M states, that's likely a DM call. But how should the DM approach?
Well, it's a beam of light that condenses into a bead. How much does a bead of light weigh? I'd think that a bead of magic light does not weigh anything, but another DM may believe otherwise. Then it's a question of does a bead of light weighs more than 1lb. Even if it did have a mass, it's unlikely to weigh more than 1lb.
Without meeting the weight requirement, you can't use Catapult to launch the Bead for extra damage (pending the save failure).
Throwing the Bead
There is another option for interacting with the Delayed Blast Fireball:
If the glowing bead is touched before the interval has expired, the creature touching it must make a Dexterity saving throw. On a failed save, the spell ends immediately, causing the bead to erupt in flame. On a successful save, the creature can throw the bead up to 40 feet.
This does imply that it can be handled, but it doesn't answer weight or anything else. Ultimately, this is a magic bead made of condensed light and it does not necessarily mean it has mass. It just means that you can throw the magic bead. If a DM does rule that it has mass, then it still needs to weigh more than 1 lb (see discussion above on weight of condensed light.)
Best Answer
Yes you can
The main question is if an container counts its content weight towards its own weight.
There is no explicit rule that says that a container is considered to count as it's weight plus its content but I would consider it logical. Especially that the exception to this have it specifically noted see Bag of holding.
Considering this I would say that the container with content can be used for this spell as long its not being worn or carried.
The second question is how the container and content are affected by the spell.
The Catapult spell has a very clear description of how the object moves:
This to me means that the object isn't just flung from it's point of origin using a ballistic trajectory. In other words the magic spell interacts with the object for the duration of its movement. This would mean the spell grabs the object and moves it along the line.
This I would say means the containers content stay in the container and impact on the target always counts as one impact.
The third question then is if i use this spell to hit a vampire with a bottle of holy water or something else with Alchemist's fire. Does is do extra damage?
My answer is no. Mostly because the spell doesn't say it does. Nor would you have it do extra damage if you used the spell to launch a sword.
As a DM I might allow you to change the damage type of the spell so the Alchemist fire would change the damage type to Fire, holy water might make it Radiant, a sword could make it Slashing and Catapulting an oil lamp would make it Fire damage. But this would be a house rule for my group.
The whole answer to question 3 is very much up for ruling by your own DM. There is no rule on how this works so your DM can rule it in any way he likes.