The hide functionality is all about hiding nodes (not objects) - please read the specifications, you can find it here: http://ez.no/community/specs/hide_unpublish
Regarding the login phenomenon: Sounds a bit strange, I'm not 100% sure if it is supposed to work like that or not. I'll ask Vadim (who made much of this neat functionality) if he has any time to answer you. Stay tuned.
"If the new ignore_visibility parameter of the fetch function is set to true, does this mean that hidden objects will be included in the fetch result?"
Yes.
"whether one can view a hidden object or not is associated with the siteaccess"
Yes, add this to your site.ini.append to be able to see hidden nodes:
[SiteAccessSettings]
ShowHiddenNodes=true
"Also, it seems that nodes are hidden, and not objects. Is it possible to hide an object and thereby also hide all of its nodes?"
No, hiding is done on per-node basis, due to speed reasons.
Please look at http://pubsvn.ez.no/viewcvs/ezp/trunk/doc/features/3.5/hiding.txt for more information.