There is no way to gain skills in Odyssey. A core concept of the Odyssey campaign is that the most important kind of progression that characters make is in their reputation and their respect or fear from others. However, you may change your choice of skills, kit or path as you wish between events.
No. However, the gods, through their favour, and philosophers, through their mysteries, may be willing to grant your character additional abilities. Rarest of rare, magical artefacts like the Helm of Athena or the Sword of Damocles exist that may grant all manner of abilities.
The kits are named for a particular archetype. However, those names are simply labels. If you wish, for example to play a Greek peltast rather than a hopilite, you’ll find the hopilite kit provides you with all the skills you might need. If none of the kit skill lists for your culture fit your concept, you can always construct your own character using the skill picks in the rulebook.
Characters may go through any gate opened by a priest of their own culture provided they are conscious and willing. Characters of any path may travel through a Gate of Horn, but warleaders are forbidden from travelling through the Gates of Ivory by edict of the gods.
Access to some areas reached by gates may still be restricted. For example the Chamber of the Great Game and the World Forge are only reached by gates that are restricted to priests and philosophers respectively.
No. Characters passing through either gate must be conscious and willing.
A referee must be present if you are travelling to see a god. A referee need not be present if you are travelling home, to the Chamber of the Great Game or to the World Forge.
No. The atmosphere of combat in the arena makes giving the direction and instruction required impossible. The supervision of attendants is akin to a junior surgeon being closely and individually supervised by a senior colleague, not the coach of a football team.
You may claim a crafting background, such as being a blacksmith or artist but this will have no in-game effect and should not overlap with the other defined skills and paths in the game.