1. An object given to a hero to see the future.
Persian king Jamshid was said to own a seven-ringed wine cup, which was filled with an elixir of immortality and allowed him to see the future.
A dangerous location in Welsh mythology is Cadair Idris in the Snowdon mountains. The location is said to look like the seat of a giant (Idris being the giant in question). Men who spend the night there are either given the gift of prophecy, blessed with the gift of poetry (which is sometimes the same thing), or go stark raving mad.
This one isn't given to a hero, sadly, but is interesting -- Tezcatlipoca, one of the four creator gods of the Aztecs, is also called "Smoking Mirror." He's related to obsidian (which is what the Aztecs used to make mirrors), but he's also often depicted with a mirror in place of the foot he lost during the creation of the world. There's an association between the mirror and seeing all of the world, or seeing the future, depending on the source.
2. An object given to a hero that allows him/her to step into a new world/realm.
I can mostly think of fairy tales that fulfill this sort of requirement, as far as objects go -- and even then, it's usually a boat or a horse or some other vehicle with the potential to cross between the world of men and the world of the fae/the dead/some particularly difficult to reach location. In "East of the Sun, West of the Moon," for example, the heroine is given a horse; in at least one of the stories in 1000 Nights and One Night, there's a flying horse that travels faster than any living creature (and may be mechanical, if I remember correctly); and one of the Thirteen Treasures of Britain (from Welsh tradition) is the chariot of Morgan Mwynfawr, which could instantly transport the driver to the location of his desire. There's also the flying carpet, which does similar things, but doesn't really cross *realms* -- just makes travel to human-world locations much faster.
3. A structure/entrance that acted as a gateway to a new world.
There are three major categories I can think of for this -- it's a trope that appears pretty frequently. In the British Isles, particularly (and probably along parts of continental Europe), fairy hills are all over the place. They're typically sites where archaeologists have found ruins of hill forts, and such, but local tradition dictates that the hills are entrances into the Other world. One of my favorite of these is Glastonbury Tor, which is topped by St. Michael's Tower and is beautifully iconic.
The second category are caves, which can connect either to the Other world in that same fairy sort of tradition, or to the land of the dead. I believe Orpheus begins his journey to Hades in a cave, and I know there are cave entrances in Ireland that have a connection to the Other world. There are urban legend sites that mention associations between caves and the opening to the world of the dead in Latin America, as well, but I don't know how good the mythological basis is for that.
The third category is just the shoreline. At World's End, in Cornwall, for example, at certain times you're supposed to be able to look out and see the drowned land of Lyonesse, apparently accessible through the mist if one could just get there. Odysseus in the Odyssey communes with the spirits of the dead, consulting with Tiresias, after having beached his vessel. Shoreline is liminal space -- space where things aren't quite one thing or another -- so the idea of the shore as a crossing point from one world into the next is a pretty common theme. (To use Jungian analysis, it's also the place where the conscious meets the unconscious!)
For a stretch, you could say that a pok-ta-pok court (the Mayan ballgame playing ground) is an entrance into the underworld, as well, given the way that the gods of the underworld tend to summon people from one of those courts. Along those lines -- as far as a stretch goes -- there are lots of man made sites, such as tholoi or some pyramids, that are built with the intention of transporting a *spirit* to the other world. The Treasury of Atreus is a very cool looking one of these.
4. An object given to a hero to help him/her find his/her way out of the labyrinth (or a general state of confusion or blindness).
The easiest of these, is, of course, Ariadne's thread, which Theseus used to find his way back out of the Labyrinth of Crete. Fairy tales offer other options, as well -- the traditional marbles, bread crumbs, or pebbles that children or heroes leave behind them to find their way back. There are also stories about various liquids enabling a person who smeared them on his eyes to see the Other world -- dragon's blood, fairy spit, etc., etc., -- but a lot of those are literary conventions, I think.
5. An object used in storytelling ceremonies that gives the holder permission to speak to the group.
This seems to me to be most commonly a Native North American tradition, and, of course, the most common item is the "talking stick," which I'm sure you're familiar with! Some things I've read posit that the "peace pipe," feathers, or even wampum belts could be used for this same function.
A few of the things that readers mentioned in comments that I hadn't: Orpheus's lyre, herms or hermae as crossing points between worlds, traditional fortune telling devices such as crystal balls and tarot cards, the Tree of the Knowledge of Good and Evil (as an effective transport from Eden to the human realm), Coyote running toward a village with a firestick in his mouth and being chased by smoke spirits, pools of water and caves of prophecy, and literary examples like the AURYN amulet from The Neverending Story, the palantiri from The Lord of the Rings, and Adaon's brooch from The Black Cauldron.
I've asked to see the eventual logo that sparked these questions for the design team in the first place, and should I be allowed to, I'll share it here. In the mean time, what fun hunting for mythic objects!