Can you throw things with MAGE hand?

I see no issues with dropping something with a mage hand cantrip. It requires an Action on behalf of its caster (bonus Action in the case of an Arcane Trickster) to drop something the hand is holding. Thus, as long as the object fits the requirements of the cantril, the Hand can drop something.

Do arcane tricksters get mage hand?

The Arcane Trickster just uses the regular Mage Hand, but they get some extra bonuses to it. That doesn’t make it a separate spell.

Can mage hand steal weapon?

No. The basic text of Mage Hand says it can manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial. It cannot attack, activate magic items, or carry more than 10 pounds.

Can you pickpocket with mage hand?

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.

What can an Arcane Trickster do with the Mage hand?

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:

Can a mage use legerdemain to steal a weapon?

This is a question about the Arcane Trickster using Mage Hand Legerdemain to steal weapons. It looks like there’s a pretty solid answer there that Mage Hand Legerdemain can be used to steal a weapon off of someone as long as it’s in a container (carried or worn).

Can a mage hand make carried objects invisible?

Carried objects becoming invisible would be an effect worth mentioning. Of significance is that the normal mage hand is spectral, but the description does not describe carried objects as being spectral. And there is no sense that the spectral hand actually has any substance to it.