Eruvin 34: Accessing the Physical Eruv (Or Not)

Do you really need to bring the eruv techumin to the place where you want to set up the eruv? (In the case where Shabbat follows yom tov) Perhaps the concept of the eruv is sufficient, without having a physical eruv set up. But the case of when yom tov falls *after* Shabbat is exactly one that poses a difficulty for that theoretical eruv approach - the theoretical has to be possible! Also, if you put an eruv deep in the ground, that makes sense if it's a private domain, which goes up to the sky, and deep into the earth. But what if the pit is in the public domain - where, then, is the temporary residence supposed to be? Above the pit - in which case the eruv is in the private domain of the pit? Or in the private domain deep in the pit - that's certainly the same domain. So the question hinges on a pit that's located in a karmelit, and where you want the residence to be above the pit. Plus: A case where the eruv is in a cupboard, and the key is lost... Is that eruv still valid? Is there another way to access the eruv?