Minecraft – How to heal with arrows

If this is possible, the player who fired the arrow, either by UUID or by username, should be stored somewhere for each arrow.

First I checked the wiki for an arrow entity's NBT data. But as you said, while ThrownEnderpearl, ThrownExpBottle, ThrownPotion, and Snowball all have an ownerName field, Arrow does seem to have anything indicating it would store the player who fired it.

I fired an arrow and checked its NBT in game, as I've noticed the wiki occasionally misses out data tags:

But again there was nothing that stored my player ID. (Note that UUIDMost and UUIDLeast are both for identifying the entity itself, not who fired it)

I wanted to see if the data of who shot an arrow persists through the world closing and opening. To do this, I:

1. Created a world
2. Shot an arrow straight up
3. Closed the world
4. Opened the world
5. Quickly opened the world to LAN
6. Logged in with an account that was on survival, positioned directly below the arrow

When the arrow hit the survival player, not only was no ping sound played for hitting a player, but the message in chat simply showed that the player was shot by "Arrow", rather than me:

(Note that both players were logged on, the one that shot the arrow and the one that got hit by it)

Because of this, I believe that the player who fired an arrow is not actually stored anywhere in an arrow's data.

Thus the answer to "Is there any way to summon arrows that are shot by a player or mob?" unfortunately seems to simply be no.

I suspect death messages and ping sounds are done temporarily in code, rather than stored anywhere that can be manipulated with commands. They are lost when the world reloads, and are not stored anywhere in the individual arrow's data.

Did he set the speed to -1? That causes the controls to become flipped. Try entering speed 1 and see if that fixes it.