Changing goal of Special Stages

Is it possible to create alternate Special Stage maps that have a different goal than collecting X rings in Y seconds? For example, could one beat a level within a certain amount of time, or beat a boss to get the Emerald? If so, how would one do that?

For example, could one beat a level within a certain amount of time

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Use Countdown = <value> in Level header.

or beat a boss to get the Emerald?

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Depends on the way you want: for instance, it's possible to make "boss defeat" trigger a linedef executor that opens a door or does another in-level execution. I don't know how, but I know the 1st boss of The Emerald Isles does something similar.

I think I didn't type clearly enough. I want to know if it's possible to complete a Special Stage by entering an Exit Sector instead of collecting a number of Rings.
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So all I have to do is insert an Exit Sector into a level, use a WAD to replace a Special Stage with it, and I'll acquire the Emerald when I touch the Exit Sector?
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Wait a minute. Did someone just delete Sky's post?

Sky deleted his own post. I dunno if you can do exactly what you're intending, but a trivial way to accomplish it in the abstract is to make the ring requirement 1, and just put the single ring at the end of the stage.