Does it heal on the caster's turn if you cast it on someone or move it on top of someone?
No. This Sage Advice from April 2016 answers why:
Does moonbeam deal damage when you cast it? What about when its effect moves onto a creature? The answer to both questions is no. Here’s some elaboration on that answer.
Some spells and other game features create an area of effect that does something when a creature enters that area for the first time on a turn or when a creature starts its turn in that area. The turn you cast such a spell, you’re primarily setting up hurt for your foes on later turns. Moonbeam, for example, creates a beam of light that can damage a creature who enters the beam or who starts its turn in the beam.
Our design intent for such spells is this: a creature enters the area of effect when the creature passes into it. Creating the area of effect on the creature or moving it onto the creature doesn’t count. If the creature is still in the area at the start of its turn, it is subjected to the area’s effect.
The spell requires the creature to move into it or start a turn there; moving the spell onto a creature or casting it on a creature's space has no effect.
If the spirits space is occupied but you use double movement to move through it can you still heal?
Yes. The spell simply requires that you enter the space at some point during your turn (or begin a turn there). There is no requirement to stay in it, so passing through it qualifies.
Out of combat can the whole party take turns moving through the spirit to heal 10d6 each, taking 10 turns moving through the spirit in turn order and everyone heals 10 times?
RAW, yes. The spell lasts for one minute, so theoretically your party could spend that 60 seconds hopping in and out of it and receiving the healing effect each time you do. However, personally, if I were your DM, I'd rule no because of the inherent sillyness of the party receiving significantly more healing as a result of spending less time in the spirit's light. Your DM may disagree, of course.