- Does it heal on the caster's turn if the space you initially cast it on is already occupied by someone, or if you move it on top of someone?
No. This official ruling from the April 2016 edition of the Sage Advice column 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 healing spirit 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 spirit's space is occupied, but a character uses double movement to move through that space, can that character still be healed?
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's space to each heal a total of 10d6 HP?
According to the errata released in 2020, no:
Healing Spirit (p. 157). The following text has been appended to the second paragraph: “The spirit can heal a number of times equal to 1 + your spellcasting ability modifier (minimum of twice). After healing that number of times, the spirit disappears.”
Healing spirit was heavily nerfed, and your proposed method no longer works.