All of the ones in the skills menu.
play skyrim
There are no classes in Skyrim, a character is free to develop and become proficient in whichever skills they wish.
Skyrim has no class system, so how you define a 'barbarian' might be different from others. The Companions in Whiterun are probably the closest to the archetypical Nordic Warriors, you'll find trainers there for various melee skills, including weapons, shields and armour.
There's a student who walks around in the collage of Winterhold. He specializes in illusion spells, and can train you, and sell you illusion spells.
Yes and the longer you stay in jail the more skills are affected.
To better use the grindstone you need better smithing skills.
Yes, there are no classes. Any character can learn spells and start developing magic skills.
Yes, as long as you use them on valid targets, it doesn't matter if you are on horseback. This applies to all combat skills.
You train you skills.
Train skills.
Train your skills.
Skyrim is a 15.