They can be cast at will.
Magic Initiate states:
You learn two cantrips of your choice from that class's spell list.
The restriction on casting once per day is placed only on the 1st level spell, in the following bullet:
In addition, choose one 1st-level spell from that same list. You learn that spell and can cast it at its lowest level. Once you cast it, ...
If the caster is on a moving platform, and is moved more than 30 feet away, Mage Hand will disappear.
It would be a DM ruling to say that the Mage Hand's position is relative to a surface, and thus moves with the platform it is on. I think that is a perfectly logical ruling to make, depending on the scenario.
The hand vanishes if it is ever more than 30 ...
This does not work
Quick Toss states (emphasis mine):
As a bonus action, you can expend one superiority die and make a ranged attack with a weapon that has the thrown property. [...]
The magic stone cantrip states:
[...] You or someone else can make a ranged spell attack with one of the pebbles by throwing it or hurling it with a sling. If thrown, it has ...
"Running water" doesn't simply mean water that is moving at the moment.
"Running water" as a term specifically refers to rivers and streams, not just water that happens to be in motion. A thrown water bottle is not 'running water', nor is a small continuous leak from a water pipe.
Today we more commonly use the term 'running water' to ...
Only if you use it to create running water
The movement of the water itself is instantaneous, so is unlikely to count as running, and more specifically the vampire has to end its turn in the running water, which this spell can't do.
However if you put the water up a height so it slowly dribbled and formed a thin stream that the vampire for some reason stood ...
No, the vampire cannot end its turn in water that is being moved by shape water.
Shape water dictates that it moves water instantaneously. This indicates that a vampire could not end its turn in water that is moving via shape water, since the water moves instantaneously. Water that moves instantaneously also cannot be said to be “running” in any meaningful ...