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.

“A wizard did it”

You’ve probably heard the phrase before to explain with a hand-wave why something weird exists, like bulettes or owlbeards: “a wizard did it.” There are things in D&D that were made by human Dungeon Masters for convenience of creating fun adventures, not via rules for players.

Like owlbears were (allegedly) created by a mad wizard, so is a lot of Undermountain.

Halaster’s apprentices are mad and powerful inventors

Halaster’s apprentices are older than Waterdeep. The city was built on the ruins of their original wizards’ tower. Today, those who still live, are ridiculously powerful archmages. Halaster himself, if he still lives (the jury is still out), is nearly a god-like force of nature in his own home.

The entire existence of Undermountain was originally to give Halaster and his apprentices a place to learn secrets of magic and develop new spells and things in private. The place is packed with bizarre, unique creations that have not been shared with wizards outside its halls.

His apprentices guard the secrets of their discoveries and creations jealously (first and foremost from each other — only secondarily do they care about the world above), and nobody knows how they do what they do.

In a less lore-focused way, Halaster and his apprentices were created to be plot devices, which could be used to explain anything non-standard that exists in Undermountain.

Every spell was invented by some wizard once

Earlier editions of D&D had spell research rules, and those are part of the lore of even 5th edition. For example, Tenser’s Disc was invented by Tenser (a PC played by Gary Gygax’s son Ernest in Gary's campaign). The spell wasn’t available until Tenser researched it and shared it with other wizards, so that knowledge of the spell spread throughout Greyhawk and, before long, across the Multiverse.

Halaster and his apprentices have invented many spells, rituals, and partly-magical inventions. You will find many unique magics under Waterdeep, but won’t necessarily learn the secrets behind them easily, if ever. Those secrets are buried deep, and they don’t like to share.

There is probably a ritual to create more of these living unseen servants, but it’s likely hidden away in the carefully guarded spellbook in the carefully guarded secret lair of a mad and dangerous archmage somewhere deep in the lower levels of Undermountain.

To read more

The original lore on Halaster and his apprentices is first established in their first published appearance, the AD&D 2nd edition megadungeon boxed set The Ruins of Undermountain. The price of the PDF is well worth it just for the lore, but it’s an excellent playground for PCs of all levels as well. (Converting it to 5e is likely a pain though, and I haven’t tried yet so can’t recommend either way.)