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.
Anything referred to as an "attack roll" is made against the AC of the target.
Whenever you make an attack roll, whether that's with a weapon or a spell, you are rolling against the Armour Class of the thing you're attacking. I don't have access to the cut-down rulebook that comes in the Lost Mine of Phandelver starter set so I cannot reference that or tell you exactly what the wording in there is, but in the Basic Rules we have these rules about making attack rolls (PHB pg 193-194):
Whether you're striking with a melee weapon, firing a weapon at range, or making an attack roll as part of a spell, an attack has a simple structure.
...
When you make an attack, your attack roll determines whether the attack hits or misses. To make an attack roll, roll a d20 and add the appropriate modifiers. If the total of the roll plus modifiers equals or exceeds the target's Armor Class (AC), the attack hits. The AC of a character is determined at character creation, whereas the AC of a monster is in its stat block.
This last part is somewhat dubious because there are lots of things that can change a character's Armour Class, like putting on or taking off armour, casting spells that affect AC, or gaining the benefit of certain magic items (options which are also often available to the monsters/NPCs that the characters will fight, too). I think the starter set you're playing probably avoids that as much as possible though to simplify character management for beginners.
Best Answer
Should rolls be made on content breakage per attempt made on the box
There is a brief description of HP for items in the DMG (and repeated in the SRD), which basically says objects have HP, when that HP is exceeded it breaks. The rule books don't have any specific rules about damage carry over.
A fair house rule where any damage beyond the chest is done to the objects (evenly dividing or most fragile first -- up to the DM). If they exhaust the objects inside, they also break.
There is also armor classes for objects (DMG 246), but if the item is immediately in front of you, and there is no threat, rolling against AC is a silly exercise, as they'll just keep rolling until they finally hit it. Object AC makes more sense if in the heat of battle (or on the run fleeing) a player wants to smash something.
Are potion bottle and scrolls generally strong enough ... ?
A fragile Chest is 3 HP, and a resilient chest is 10 HP It also suggests a fragile Potion bottle is 2 HP, and resilient bottle is 5HP.
This chart is on 247 of the DMG.