From the PHB (emphasis mine):
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 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.
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.
And the invisibility spell (emphasis mine):
A creature you touch becomes invisible until the spell ends. Anything the target is wearing or carrying is invisible as long as it is on the target’s person. The spell ends for a target that attacks or casts a spell.
I know the spell is slightly different but it’s the most comparable thing I could find.
I lift some jail keys off of a guard and then turn the hand invisible.
I grab 3 platinum coins off the mayor’s desk when I think no one is watching and close the hand around them so they’re totally inside the fist.
I give the hand a lockpick and pick a lock from across a crowded room.
In which of these situations (if any) would the objects be visible?
Context of the question: I'm the Rogue player, not the not DM. I view the mage hand for the arcane trickster as a type of highly specialized familiar because I can summon or dismiss it a will, and I can "command" it to do some simple tasks. But I'm fairly new and inexperienced so I might be looking at it wrong. My reason for posting the question was to see if I am missing something that should have been obvious, or if there was room to play around with this class feature.