Minecraft – Activate piston on command

The chain command blocks have to be set as "Always active" (you just click the "Needs redstone") or apply redstone signal.

or you simply can’t use exclamation marks in NBT tags

This is the case unfortunately. Using ! for negation only works in selectors, and even then only for certain arguments (type, name, tag, team, m).

The problem is, that when I move far away from the setup, the redstone stops working, leaving me unable to summon any new fireballs. Instead, I'm trying to use a command block to check when anything except for item is being held;

If I'm understanding correctly, this problem won't be solved by changing your commands if they already do the correct thing when you're near them.

You should instead move your contraption to spawn chunks (which are always loaded) or keep the chunks your contraption is in loaded with a chunk loader.

how I can check for anything except for a specific item?

Your original commands will leave players not holding the item with a score of 0, meaning you can do something like:

/scoreboard players add @a Fireball 1 {SelectedItem:{tag:{display:{Name:"Summon Fireball"}}}}
/execute @a[score_Fireball_min=1,score_Fireball=1] ~ ~ ~ /summon Fireball ~ ~2.5 ~ {direction:[0.0,0.0,0.0]}
/execute @a[score_Fireball_min=0,score_Fireball=0] ~ ~ ~ /say I'm not holding the fireball summon item
/scoreboard players set @a Fireball 0

You can rearrange these commands slightly to have players be left with 0 or 1 at the end of the tick depending on whether they're holding the fireball, if you want:

/scoreboard players set @a Fireball 0
/scoreboard players add @a Fireball 1 {SelectedItem:{tag:{display:{Name:"Summon Fireball"}}}}
/execute @a[score_Fireball_min=1,score_Fireball=1] ~ ~ ~ /summon Fireball ~ ~2.5 ~ {direction:[0.0,0.0,0.0]}
/execute @a[score_Fireball_min=0,score_Fireball=0] ~ ~ ~ /say I'm not holding the fireball summon item

Further, I’ve tried using exclamation marks to set the score of all entities, including the ones with a blank score, but except for the ones with a score of 2 or higher, to 0;

The score_... argument also unfortunately doesn't accept ! for negation. You'll instead need to do this in multiple steps:

1. Add 0 to all scores, turning null scores to 0: /scoreboard players add @e[type=Arrow] Arrows 0
2. Do the command for arrows with Arrows score less than 2: @e[score_Arrows=1]
3. Do the command for arrows with Arrows score more than 2: @e[score_Arrows_min=3]