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.

The rules, as written, don't say.

There are no specific rules regarding the contents of an invisible mage hand but there is evidence that objects held in the hand may be invisible.

While we don't have many general rules regarding invisibility (mainly just what's in the "Hidden" sidebar in the PHB), we do have an invisibility spell and an invisibility monster action to base our decision off of. We also have usability to consider; why would the designers give us the ability to turn the mage hand invisible if it gives away its position as soon as it picks something (anything) up?

Invisibility the spell says that all objects worn or carried by the target are invisible. This seems fairly harmless and renders the spell kind of useless if it doesn't give completely invisibility to all objects carried and worn.

Under the control of a skilled arcane trickster, the mage hand can be used to manipulate objects, pick locks, pick pockets, or carry an object or objects weighing up to 10 lbs. It can also turn invisible, per the arcane trickster's class feature.

With that in mind, we can probably make some assumptions about the hand:

• It is probably the size of an average person's hand
• It is dexterous, or at least as dexterous as its controller
• It can manipulate small objects such as a lockpick
• It can steal things from people

We also know the following:

• In other cases in the rules, invisibility turns held objects invisible. This alone is not reason enough, but because it makes the spell invisibility useful (otherwise enemies would see your sword and attack you sight unseen) it is important to the argument (usability, as I mentioned above).
• The arcane trickster is a class archetype based around a rogue who uses magic to play tricks and get away with his roguish activities.
• The arcane trickster gets special rules for his mage hand.

Taking these things into consideration, it seems quite in line with the design goals of 5e and the overall flavor of the class to allow objects held in the hand and completely obscured by it to become invisible. This includes as many coins as can fit inside the closed fist, a dart, dice, a key, lockpicks, or other small, "palmable" objects. Larger objects, such as weapons, planks of wood, mugs of ale, and so on, would be visible because they are not completely obscured by the mage hand.

My reasoning for this:

• Invisibility the spell and invisibility the monster action generally grant invisibility to the target and all items worn or held.
• The arcane trickster relies on subterfuge and trickery. Why give them an invisible mage hand if it can't hide anything inside it? That's like saying, "Here's a beer, but don't drink it!" Or maybe more like, "Have this beer, you can drink it, but only when no one is looking. Oh yes and you're in a crowded marketplace."
• For the skeptical DM: It doesn't hurt anything to allow this! In fact, it will probably make your games more fun when your Arcane Trickster's player is having more fun.
• It isn't broken. It doesn't imbalance anything in the gameplay and it's easily overcome or made up for in other areas by a good DM. Personally, I don't think it even needs to be "made up" for. It is by no means a showstopper.