Apoxie sticks to everything that's not wet, pretty much, so it's advisable to wet your tools and use gloves when working with it. And it sticks very well if the two parts are mixed correctly, I have had little apoxie bits stuck to my floor for months now.
Yup, wet paperclay sticks to dry paperclay. All in all, paperclay acts somewhat close to how waterbased clay (the "natural" kind) acts: it's air-drying tho waterbased clay needs to be fired whereas paperclay doesn't, it can be modelled similarly and dry(ing) clay can be revived with small amounts water to get it supple again as long as it's not too dry. Adding wet clay to dried clay works similarly too, you just wet the drier surface a little to make sure the fresh clay sticks real well.
You don't have to use a base shape for paperclay, but a sturdy base, cardstock etc, can be useful if you want the bottom of the piece to remain flat and not warp, in case you need to move it around while it's drying. Using only paperclay works too.
You can also use the styrofoam shape and remove it, the sturdiness depends on how thick the paperclay dome is. It has to be fully dry when you remove it to ensure it doesn't warp, so wait around 24 hours before removing it.
Paperclay is pretty sturdy, it may crack and usually has little cracks after sanding which can be filled with fresh paper clay or just painted/glue-coated over. I have dropped the mask I mentioned a couple of times and it's in perfect shape, at least on the surface.
Here's a couple of useful tutorials on paperclay:
(step 5 onward, introduces the material pretty well, good tips)
(this uses a polystyrene core, seems to work well)
Anyway, the pva (white) glue sealing for paperclay I mentioned is optional, seems that people have gotten good results with just paint (primer first). But I like the sturdiness the glue seems to give, it feels less prone to cracking somehow.