The Stack, Ability Resolution, and Checking Conditions
Anyway, ranting aside, let's look at some examples.
Spellstutter Sprite has the following ability:
When Spellstutter Sprite enters the battlefield, counter target spell with converted mana cost X or less, where X is the number of Faeries you control.
So, let's say my opponent controls a Vendilion Clique. I decide that I'm going to cast a Tarmogoyf, and he casts Spellstutter Sprite in response. At this point, the stack looks like this:
Spellstutter Sprite
Tarmogoyf
The Sprite resolves first, and it's ability goes on the stack, with X=2 since both the Sprite and Clique are on the battlefield. The stack now looks like this:
Sprite's ability, X=2 targeting Goyf
Tarmogoyf
At this point, the Goyf is a legal target to counter since it's converted mana cost (CMC) is 2. However, I decide to Terminate his Clique in response. The stack now looks like this:
Terminate
Sprite's ability, X=2 targeting Goyf
Tarmogoyf
My Terminate resolves, killing the Clique. The stack is:
Sprite's ability, X=2 targeting Goyf
Tarmogoyf
It's now time for the Sprite's ability to resolve, however, it's the only Faerie on the battlefield. At this point, the ability checks its conditions again, and notices there's only one Faerie, which makes X=1. At this point, Tarmogoyf is no longer a legal target for the ability. The ability itself is countered, and Goyf resolves.
For more on this, Natedogg posted the FAQ at http://community.wizards.com/go/thread/view/75842/18826866/?pg=last.
Now, let's look at another example.
Vedalken Shackles has the following ability:
Gain control of target creature with power less than or equal to the number of Islands you control for as long as Vedalken Shackles remains tapped.
Let's say I have a Dark Confidant on the battlefield. My opponent activates his Shackles, controlling 3 islands, targeting my Confidant, which has power 2. I respond with Jund Charm, putting two +1/+1 counters on my Confidant. At this point, the stack is:
Charm -> +1/+1 counters
Shackles -> Gain control of Confidant, islands = 3
My Charm resolves first, making my Confidant a 4/3. That leaves the ability of Shackles on the stack. However, my opponent has a Misty Rainforest on the battlefield, which he activates. The stack is now:
Rainforest -> search for a Forest or Island
Shackles -> Gain control of Confidant, islands = 3
My opponent fetches an Island and puts it on the battlefield, upping him to 4 islands. The ability of Shackles is next to resolve. At this point, Confidant's power is 4, but my opponent also controls 4 islands, meaning Confidant remains a legal target. (Don't worry, I had an Ancient Grudge in hand too )
In both cases, the conditions for an ability on the stack changed upon resolution. In one case, it countered the ability, but in the other, it was still valid, but only after my opponent made a play to keep it valid.
So, the next time a triggered or activated ability is played, remember that it goes on the stack first, and makes sure any targets of it are legal both when it's put on the stack and when it resolves. It may just make the difference between winning and losing.
The judge was indeed wrong and doesn't understand how the stack works.
In the situation you mentioned, your opponent can tap land targeting and add red, or even float it for that phase (remember mana empties between phases). At that point, Karn's ability has been added to the stack. If your opponent elects to cast a bolt, the bolt is on top of the stack (last in first out).
At that point, Bolt resolves first, killing Karn. However, that doesn't remove Karn's ability from the stack. All targets for the ability are still legal (the Steam Vents) and so the ability resolves because checking on resolution passes. The target hasn't been altered at all; it was and still is the Steam Vents. Karn doesn't need to be present for the ability to resolve.
Your opponent can keep that red mana floating until that phase ends. Once you enter either combat or the end phase it's gone though.
A simliar situation is true for abilities when creatures enter the battlefield. Say you play Vendilion Clique on your main phase. Clique's ETB ability goes on the stack. Your opponent may Bolt the Clique in response to the ability, but even with Clique gone the ability still resolves.