Storage was indoors at the Tomorrowland terminal. I remember peeking in there while waiting in the queue.
Cabins were pushed thru the stations by muscle power. CM's brought the cabins to a full stop for both unloading and loading. I'm pretty sure there was no raised platform, so boarding while in motion would have been difficult for many Guests. After loading, cabins were pushed onto a declining track to reattach.
Here's an image showing the cabins stacked up in the load area. If it was not a detachable system, they would be spaced as they were on the cable.
https://vignette.wikia.nocookie.net...neyland.jpg/revision/latest?cb=20120201075136
(Sorry, looks like that image was actually of the Anaheim installation.)