The PHB page 277 under "Stunned" reads:
"You can’t take actions"
So if the screen says differently, there must be some RAW/RAI mixup here. I've been ruling that you can't take actions, so you can't speak or drop a held item to the ground for example. My players spend their stunned turns glaring very angrily and thinking mean thoughts. Those are apparently not actions.

While on this subject, what about multiple effects of the same time that do *not* stack; do they persist but don't stack (meaning that each requires a save) or do they overwrite the previous effect (meaning only one save).

For example, Jim the Paladin gets attacked by three zombies that deal 5 ongoing necrotic damage. I know that Jim will not be taking 15 ongoing damage as ongoing of the same type will not stack, but is Jim under the effect of three ongoing 5 necrotic damage effects (each requiring a save to clear) or does Jim only have to make one save to end the effect?

Yes, jim has to save vs. each effect, and if they happen to hit him again, the effects will stack.

Thanks! Exactly what I was looking for, and also how I went with it ingame

To further extend the save discussion, if Jim the Paladin is under the effect of two different sources of Immobilization applied by two different enemies, is he under one Immobilization requiring one save or two different effects each requiring a save? What if he is hit by an attack that causes Immobilization that has been imbued with a poison causing Immobilization?

Based upon the earlier responses, I'm assuming that he is rolling a save versus each effect.

Which means that the Ranger attack powers might not be as redundant as they initially appear.

Note that the FAQ gives just a suggestion - its entirely RAW to have the illusions spread out in 5' radius around the caster, and its only for ease of use that it suggests placing all of them in one square. If a player wants to spread out his illusions for greater tactical advantage, its entirely within the rules to do so.

Though, common sense dictates that if you have enough illusions, they must spread out to other squares some...5 illusions of a human in a 5' square is just silly.

Ongoing damage effects never stack unless explicitly stated that they do (and I don't know of any such effect.)

I'm pretty sure that identical effects from identical sources do not "multiply", that is, they only require one saving throw - I would count identical monsters as identical sources for this purpose. So if you were slowed by an Aboleth Slime Mage's Slime Orb and also by an Ice Archon Rimehammer's Maul, you would have to save against each effect separately. But if you are slowed by the bites of two Clay Scouts, you only need to save once.

That's how I would rule, but that is not RAW. The general rule is:
"Save Ends: The effect ends when the target makes a successful saving throw against it."
"Overlapping Durations: If a target is affected by multiple powers that have the same effect but end at different times, the effect with the most time remaining applies."
Implying that if it's the same effect from different sources, you still only need to save once.

As for (save ends both) I found a better written rules text - see page 279 under Saving throws:
"Another kind of effect is like an imp’s hellish poison, which includes both ongoing poison damage and a –2 penalty to Will defense. You
don’t make separate saving throws against the ongoing poison damage and the Will defense penalty; you make a single saving throw each round against the hellish poison itself.
Some powers create effects that require multiple saving throws to fully escape. These powers include aftereffects that apply after you save against the initial effect. For example, a power might knock you unconscious until you save but have an aftereffect that slows
you. Once you save against the unconscious condition, you need to save against the slowed condition before you’ve fully escaped the power’s effects. An aftereffect doesn’t begin until after you’ve rolled all your saving throws at the end of your turn."

sdorry, by Stack I mean you'd have to require another save.

I believe different attack = different source thus the same monster could slow you several times and you'd have to make different saves.

Naturally
The original question was re: Images and AoOs, so for that situation all in one place is a good way to rule it rule-wise. Actual game/display-wise, I spread em out as appropriate

Instead of just saying "You could believe that, but you'd be wrong" and sounding like a jerk, I'll ask:
If an identical creature using an identical attack to apply an identical condition with an identical duration is a different source, then what would be the same source to you? A power that reads "Target is slowed twice (save ends both)"?